Covering gonality of symmetric products of curves and Cayley–Bacharach condition on Grassmannians

Abstract

Given an irreducible projective variety X, the covering gonality of X is the least gonality of an irreducible curve $E\subset X$ passing through a general point of X. In this paper, we study the covering gonality of the k-fold symmetric product $C^{(k)}$ of a smooth complex projective curve C of genus $g\geq k+1$. It follows from a previous work of the first author that the covering gonality of the second symmetric product of C equals the gonality of C. Using a similar approach, we prove the same for the $3$-fold and the $4$-fold symmetric product of C.

A crucial point in the proof is the study of the Cayley–Bacharach condition on Grassmannians. In particular, we describe the geometry of linear subspaces of $\mathbb {P}^n$ satisfying this condition, and we prove a result bounding the dimension of their linear span.

1 Introduction

In recent years, there has been a considerable amount of interest and work concerning covering gonality and, more generally, measures of irrationality of projective varieties (see, for example, [5, 6, 9, 11, 16, 20, 26]). Along these lines, we are interested in determining the covering gonality of symmetric products of curves. To this aim, we also come across the study of the Cayley–Bacharach condition on Grassmannians, a topic of independent interest, and we describe the geometry of linear subspaces of $\mathbb {P}^n$ satisfying this condition.

Let X be an irreducible complex projective variety. We recall that the gonality of an irreducible complex projective curve E, denoted by $\operatorname {\mathrm {gon}}(E)$ , is the least degree of a non-constant morphism $\widetilde {E}\longrightarrow \mathbb {P}^1$ , where $\widetilde {E}$ is the normalization of E. The covering gonality of X is the birational invariant defined as

$$ \begin{align*} \operatorname{\mathrm{cov.gon}}(X):=\min\left\{d\in \mathbb{N}\left| \begin{array}{l} \text{Given a general point }x\in X,\,\exists\text{ an irreducible}\\ \text{curve } E\subseteq X \text{ such that }x\in E \text{ and }\operatorname{\mathrm{gon}}(E)=d \end{array}\right.\!\!\right\}. \end{align*} $$

In particular, if X is a curve, then $\operatorname {\mathrm {cov.gon}}(X)$ equals $\operatorname {\mathrm {gon}}(X)$ . Furthermore, it is worth noticing that $\operatorname {\mathrm {cov.gon}}(X)=1$ if and only if X is uniruled. Thus, the covering gonality of X can be viewed as a measure of the failure of X to be uniruled. We aim to compute this invariant when X is the k-fold symmetric product of a smooth curve.

Let C be a smooth complex projective curve of genus g and let k be a positive integer. The k-fold symmetric product of C is the smooth projective variety $C^{(k)}$ parameterizing unordered k-tuples $p_1+\dots +p_k$ of points of C. This is an important variety reflecting the geometry of C and is involved in many fundamental results on algebraic curves, as, for example, in Brill–Noether theory.

We note that $C^{(k)}$ is covered by copies of C of the form $C_P:=\left\{\left.P+q\in C^{(k)}\right|q\in C\right\}$ , where $P=p_1+\dots +p_{k-1}\in C^{(k-1)}$ is a fixed point. Hence, we deduce the obvious bound

(1.1) $$ \begin{align} \operatorname{\mathrm{cov.gon}}\big(C^{(k)}\big)\leq\operatorname{\mathrm{gon}}(C). \end{align} $$

Therefore, the main issue is bounding the covering gonality from below. In [3], the first author proved that the covering gonality of $C^{(2)}$ equals the gonality of C (i.e., (1.1) is actually an equality, provided that $g\geq 3$ ).

In this paper, we prove the same for the $3$ -fold and the $4$ -fold symmetric product of a curve.

Theorem 1.1. Let $k\in \{3,4\}$ and let C be a smooth complex projective curve of genus $g\geq k+1$ . Then the covering gonality of the k-fold symmetric product of C is

(1.2) $$ \begin{align} \operatorname{\mathrm{cov.gon}}\big(C^{(k)}\big)=\operatorname{\mathrm{gon}}(C), \end{align} $$

provided that $\big (k,g,\operatorname {\mathrm {gon}}(C)\big )\not \in \left\{ (3,4,3),(4,5,4)\right\}$ .

Furthermore, a problem which naturally arises from [3, Theorem 1.6] and Theorem 1.1 is characterizing families of irreducible curves on $C^{(k)}$ which compute its covering gonality. This is, in fact, the purpose of [7], where we use similar techniques to show that if $2\leq k\leq 4$ and C is a sufficiently general curve of genus $g\geq k+4$ , then the curves $C_P\subset C^{(k)}$ defined above are the only irreducible curves covering $C^{(k)}$ and having the same gonality as C.

Concerning the exceptional cases $(k,g,\operatorname {\mathrm {gon}}(C))\in \{(3,4,3),(4,5,4)\}$ in the statement of Theorem 1.1, we cannot conclude whether the covering gonality of the symmetric product equals $\operatorname {\mathrm {gon}}(C)$ or $\operatorname {\mathrm {gon}}(C)-1$ . Moreover, we discuss the cases of low genus in Remark 3.8.

In light of [3, Theorem 1.6] and Theorem 1.1, which describe $\operatorname {\mathrm {cov.gon}}(C^{(k)})$ when $g\geq k+1$ and $k=2,3,4$ , it seems plausible that the equality $\operatorname {\mathrm {cov.gon}}\big (C^{(k)}\big )=\operatorname {\mathrm {gon}}(C)$ might hold for any $2\leq k\leq g-1$ . However, we note that if $k\geq g+1$ , the k-fold symmetric product of C is covered by rational curves, so that $\operatorname {\mathrm {cov.gon}}(C^{(k)})=1$ (cf. Remark 3.8). Finally, the remaining case $k=g$ is quite intriguing, since $C^{(g)}$ is birational to the Jacobian variety of C. In this setting, inequality (1.1) fails to be an equality for some special curves, but the behavior of $\operatorname {\mathrm {cov.gon}}(C^{(g)})$ for low genus suggests that equality (1.2) might hold when C has very general moduli (see Remark 3.8).

In order to prove Theorem 1.1, we follow the same strategy as in [3]. In particular, let $\phi \colon C\longrightarrow \mathbb {P}^{g-1}$ be the canonical map and let $\gamma \colon C^{(k)}\dashrightarrow \mathbb {G}(k-1,g-1)$ be the Gauss map of $C^{(k)}$ , which sends a general point $P=p_1+\dots +p_{k}\in C^{(k)}$ to the point of the Grassmannian parameterizing the $(k-1)$ -plane $\operatorname {\mathrm {Span}}\left(\phi (p_1),\dots ,\phi (p_k)\right)\subset \mathbb {P}^{g-1}$ . Suppose $E\subset C^{(k)}$ is a d-gonal curve passing through a general point of $C^{(k)}$ , and let $P_1,\dots ,P_d\in E$ be the points corresponding to a general fiber of the d-gonal map $\widetilde {E}\longrightarrow \mathbb {P}^1$ . According to the framework of [3, Section 4], the points $\gamma \left(P_1\right),\ldots , \gamma \left(P_d\right)\in \mathbb {G}(k-1,g-1)$ satisfy a condition of Cayley–Bacharach type, which we describe below (see also Theorem 3.2). Then a crucial point in the proof is showing that the corresponding $(k-1)$ -planes in $\mathbb {P}^{g-1}$ span a linear space of sufficiently small dimension. In particular, this governs the dimension of the linear series on C defined by the points of $\phi (C)$ supporting $P_1,\dots ,P_d\in C^{(k)}$ . We use standard results in Brill–Noether theory about the existence of linear series on C to eventually conclude that $d\geq \operatorname {\mathrm {gon}}(C)$ .

Unfortunately, the combinatorics of this approach does not work for $k\geq 5$ .

Turning to the Cayley–Bacharach condition, we consider a finite set of points $\Gamma \subset \mathbb {P}^n$ and a positive integer r. We recall that $\Gamma $ satisfies the Cayley–Bacharach condition with respect to hypersurfaces of degree r if any hypersurface passing through all but one point of $\Gamma $ passes through the last point too. This is a very classical property, whose history goes back even to ancient geometry (see [14] for a detailed treatise on this topic). The Cayley–Bacharach condition has also been studied in recent years, and it has been applied to several issues, concerning, for example, Brill–Noether theory of space curves and measures of irrationality of algebraic varieties (cf. [18, 4, 17, 22, 25]).

More generally, we consider the Grassmannian $\mathbb {G}=\mathbb {G}(k-1,n)$ parameterizing $(k-1)$ -dimensional linear subspaces of $\mathbb {P}^n$ , and we say that a finite set of points $\Gamma \subset \mathbb {G}$ satisfies the Cayley–Bacharach condition with respect to $\left|\mathcal {O}_{\mathbb {G}}(r)\right|$ if any effective divisor of $\left|\mathcal {O}_{\mathbb {G}}(r)\right|$ passing through all but one point of $\Gamma $ passes through the last point too. Of course, the case $k=1$ recovers the former definition for points in $\mathbb {P}^n$ .

We point out further that when $r=1$ , if $\Gamma =\left\{[\Lambda _1],\dots ,[\Lambda _d]\right\}\subset \mathbb {G}$ satisfies the Cayley–Bacharach condition with respect to $\left|\mathcal {O}_{\mathbb {G}}(1)\right|$ , then any $(n-k)$ -plane $L\subset \mathbb {P}^n$ intersecting all but one $(k-1)$ -plane parameterized by $\Gamma $ must intersect the last one too (cf. Remark 2.2). In this case, we say that the $(k-1)$ -planes $\Lambda _1,\dots ,\Lambda _d\subset \mathbb {P}^n$ are in special position (with respect to $(n-k)$ -planes); see Definition 2.1. It is worth noting that the study of the geometry of lines in special position was crucial in [3] and [15], in order to describe measures of irrationality of $C^{(2)}$ and of the Fano surface of cubic threefolds, respectively.

In this paper, we improve the results in [3, Section 2] on the linear span of $(k-1)$ -planes in special position. In particular, we prove the following (cf. Theorem 2.5).

Theorem 1.2. Let $\Gamma =\left\{\Lambda _1,\dots ,\Lambda _d\right\}\subset \mathbb {P}^{n}$ be a set of distinct $(k-1)$ -dimensional linear subspaces in special position with respect to $(n-k)$ -planes. Assume further that $\Gamma $ does not admit a partition in subsets whose elements are still in special position. Then

$$ \begin{align*}\dim\operatorname{\mathrm{Span}}\left(\Lambda_1,\dots,\Lambda_d\right)\leq d+k-3. \end{align*} $$

In particular, the theorem highlights that being in special position imposes strong restrictions to the geometry of the k-planes $\Lambda _1,\dots ,\Lambda _d$ since d linear spaces of dimension k in general position span a linear space of dimension $d(k+1)-1$ . We refer to Theorem 2.5 and Corollary 2.6 for more general statements, where the $(k-1)$ -planes are allowed to coincide and $\Gamma $ may admit partitions as above.

Apart from the dimension of the span of linear spaces in special position, there are other interesting issues concerning the Cayley–Bacharach property on Grassmannians and its applications; we briefly mention them in §2.3.

The proof of Theorem 1.2 is rather long and relies on various simple properties of linear spaces in special position. The idea is to consider the images of the planes $\Lambda _1,\dots ,\Lambda _{d-1}$ under the projection $\pi _p\colon \mathbb {P}^n\dashrightarrow \mathbb {P}^{n-1}$ from a general point $p\in \Lambda _d$ . If $\Gamma '=\left\{\pi _p(\Lambda _1),\dots ,\pi _p(\Lambda _{d-1})\right\}\subset \mathbb {P}^{n-1}$ does not admit a partition in subsets of $(k-1)$ -planes in special position, the assertion follows easily by induction. It takes a great deal of work to prove the assertion when $\Gamma '$ admits such a partition.

The rest of the paper consists of two parts. In Section 2, we discuss the Cayley–Bacharach condition on Grassmannians, and we prove our results on linear subspaces of $\mathbb {P}^n$ in special positions.

In Section 3, we are instead concerned with the covering gonality of $C^{(k)}$ . Initially, we recall preliminary notions on covering gonality, and we recollect the framework of [3]. Then we prove Theorem 1.1.

Notation

We work throughout over the field $\mathbb {C}$ of complex numbers. By variety we mean a complete reduced algebraic variety X, and by curve we mean a variety of dimension 1. When we speak of a smooth curve, we always implicitly assume it to be irreducible. We say that a property holds for a general (resp. very general) point ${x\in X}$ if it holds on a Zariski open nonempty subset of X (resp. on the complement of the countable union of proper subvarieties of X).

2 Cayley–Bacharach condition on Grassmannians

In this section, we are interested in the Cayley–Bacharach condition on Grassmannians. In particular, we discuss linear subspaces of $\mathbb {P}^n$ in special position in §2.1, and we prove our results on the dimension of their linear span in §2.2. Finally, we present in §2.3 further remarks and possible developments about the Cayley–Bacharach condition on Grassmannians.

2.1 Linear subspaces of $\mathbb {P}^n$ in special position

Let us fix two integers $1\leq k\leq n$ . In this section, we are concerned with collections of $(k-1)$ -dimensional linear spaces in $\mathbb {P}^n$ satisfying a property of Cayley-Bacharach type, defined as follows (cf. [3, Definition 3.2]).

Definition 2.1. Let $\Lambda _1,\dots ,\Lambda _d\subset \mathbb {P}^{n}$ be linear subspaces of dimension $k-1$ . We say that $\Lambda _1,\dots ,\Lambda _d$ are in special position with respect to $(n-k)$ -planes, or just $\operatorname {\mathrm {SP}}(n-k)$ , if for any $j=1,\dots ,d$ and for any $(n-k)$ -plane $L\subset \mathbb {P}^n$ intersecting $\Lambda _1,\dots ,\widehat {\Lambda _j},\dots ,\Lambda _d$ , we have that L meets $\Lambda _j$ too.

Along the same lines as [3, Section 3], we aim to bound the dimension of the linear span of collections of $(k-1)$ -planes satisfying property $\operatorname {\mathrm {SP}}(n-k)$ .

Remark 2.2. The connection between special position and the Cayley-Bacharach property may be expressed as follows. We recall that, given a complete linear series $\mathcal {D}$ on a variety X, we say that d points $P_1,\dots ,P_d\in X$ satisfy the Cayley-Bacharach condition with respect to $\mathcal {D}$ if for any $j=1,\dots ,d$ and for any effective divisor $D\subset \mathcal {D}$ passing through $P_1,\dots ,\widehat {P_j},\dots ,P_d$ , we have $P_j\in D$ too.

When $X=\mathbb {G}(k-1,n)$ is the Grassmannian of $(k-1)$ -planes in $\mathbb {P}^n$ , for any $(n-k)$ -dimensional linear subspace $L\subset \mathbb {P}^n$ , the Schubert cycle $\sigma _1(L):=\left\{[\Lambda ]\in X|\Lambda \cap L\neq \emptyset \right\}$ is an effective divisor of $\left|\mathcal {O}_X(1)\right|$ . Thus, if $[\Lambda _1],\dots ,[\Lambda _d]\in X$ is a collection of points satisfying the Cayley-Bacharach condition with respect to $\left|\mathcal {O}_X(1)\right|$ , then the corresponding $(k-1)$ -planes $\Lambda _1,\dots ,\Lambda _d\subset \mathbb {P}^{n}$ are $\operatorname {\mathrm {SP}}(n-k)$ .

Furthermore, if the collection $[\Lambda _1],\dots ,[\Lambda _d]\in X$ satisfies the Cayley-Bacharach condition with respect to $\left|\mathcal {O}_X(r)\right|$ for some $r\geq 1$ , the same holds with respect to $\left|\mathcal {O}_X(s)\right|$ for any $1\leq s\leq r$ and, in particular, the $(k-1)$ -planes $\Lambda _1,\dots ,\Lambda _d\subset \mathbb {P}^{n}$ are $\operatorname {\mathrm {SP}}(n-k)$ .

Remark 2.3. We note that if the $(k-1)$ -planes $\Lambda _1,\dots ,\Lambda _d$ are $\operatorname {\mathrm {SP}}(n-k)$ , then $d\geq 2$ . Moreover, the linear spaces $\Lambda _1,\dots ,\Lambda _d$ are not necessarily distinct. In particular, two $(k-1)$ -planes $\Lambda _1,\Lambda _2$ are $\operatorname {\mathrm {SP}}(n-k)$ if and only if they coincide.

Furthermore, if $\Lambda _1,\dots ,\Lambda _d$ and $\Lambda ^{\prime }_{1},\dots ,\Lambda ^{\prime }_e$ are two collections of $(k-1)$ -planes that satisfy $\operatorname {\mathrm {SP}}(n-k)$ , then $\Lambda _1,\dots ,\Lambda _d,\Lambda ^{\prime }_{1},\dots ,\Lambda ^{\prime }_e$ are $\operatorname {\mathrm {SP}}(n-k)$ .

We are now interested in sequences of $(k-1)$ -planes satisfying property $\operatorname {\mathrm {SP}}(n-k)$ that cannot be partitioned in subsequences of $(k-1)$ -planes that satisfy $\operatorname {\mathrm {SP}}(n-k)$ .

Definition 2.4. Let $\Lambda _1,\dots ,\Lambda _d\subset \mathbb {P}^{n}$ be linear subspaces of dimension $k-1$ in special position with respect to $(n-k)$ -planes. We say that the sequence $\Lambda _1,\dots ,\Lambda _d$ is decomposable if there exists a partition of the set of indices $\{1,\dots ,d\}$ , where each part $\{i_1,\dots ,i_t\}$ is such that the corresponding $(k-1)$ -planes $\Lambda _{i_1},\dots ,\Lambda _{i_t}$ are $\operatorname {\mathrm {SP}}(n-k)$ . Otherwise, we say that the sequence is indecomposable.

Then we can state the main result of this section, which bounds the dimension of the linear span of an indecomposable sequence of $(k-1)$ -planes satisfying property $\operatorname {\mathrm {SP}}(n-k)$ .

Theorem 2.5. Let $\Lambda _1,\dots ,\Lambda _d\subset \mathbb {P}^{n}$ be an indecomposable sequence of linear subspaces of dimension $k-1$ in special position with respect to $(n-k)$ -planes. Then

$$ \begin{align*}\dim\operatorname{\mathrm{Span}}\left(\Lambda_1,\dots,\Lambda_d\right)\leq d+k-3. \end{align*} $$

Before proving the theorem, we deduce the following corollary, which refines the bound given in [3, Theorem 3.3] on the dimension of the linear span of a sequence of $(k-1)$ -planes satisfying $\operatorname {\mathrm {SP}}(n-k)$ . Furthermore, this result shall play a crucial role in computing the covering gonality of symmetric products of curves.

Corollary 2.6. Let $\Lambda _1,\dots ,\Lambda _d\subset \mathbb {P}^{n}$ be linear subspaces of dimension $k-1$ in special position with respect to $(n-k)$ -planes. Consider a partition of the set of indices $\{1,\dots ,d\}$ such that each part $\{i_1,\dots ,i_t\}$ corresponds to an indecomposable sequence $\Lambda _{i_1},\dots ,\Lambda _{i_t}$ of $(k-1)$ -planes that satisfy $\operatorname {\mathrm {SP}}(n-k)$ . Denoting by m the number of parts of the partition, we have that

$$ \begin{align*}\dim\operatorname{\mathrm{Span}}\left(\Lambda_1,\dots,\Lambda_d\right)\leq d+k-3+(m-1)(k-2). \end{align*} $$

Proof. For any $i=1,\dots m$ , let $d_i$ be the number of integers contained in the i-th part, so that $d_1+\dots +d_m=d$ . Since any part indexes an indecomposable sequence of $d_i$ linear spaces satisfying $\operatorname {\mathrm {SP}}(n-k)$ , Theorem 2.5 ensures that the span of those linear spaces has dimension at most $d_i+k-3$ . Thus,

$$ \begin{align*}\dim\operatorname{\mathrm{Span}}\left(\Lambda_{1},\dots,\Lambda_{d}\right) \leq \left(d_1+k-3\right) + \sum_{i=2}^m \left(d_i+k-2\right)= d+k-3+(m-1)(k-2),\end{align*} $$

as claimed.

Remark 2.7. A partition as in Corollary 2.6 always exists. In particular, if the sequence is indecomposable, we have the trivial partition $\{1,\dots ,d\}$ , and we recover Theorem 2.5. Furthermore, the partition is not necessarily unique, and the best bound is given by partitions having the least number of parts.

Remark 2.8. If $k\geq 2$ and $\Lambda _1,\dots ,\Lambda _d\subset \mathbb {P}^{n}$ are $(k-1)$ -planes that satisfy $\operatorname {\mathrm {SP}}(n-k)$ , then [3, Theorem 3.3] asserts that $\dim \operatorname {\mathrm {Span}}(\Lambda _1,\dots ,\Lambda _d)\leq \lfloor \frac {kd}{2}\rfloor -1$ . Thus, Theorem 2.5 gives a strong improvement of this bound when the sequence $\Lambda _1,\dots ,\Lambda _d$ is indecomposable and $k\geq 3$ .

It is also easy to see that Corollary 2.6 improves [3, Theorem 3.3], apart from the case $k=2$ and for few exceptional configurations of the $(k-1)$ -planes $\Lambda _i$ , where the two bounds coincide. Moreover, both Theorem 2.5 and Corollary 2.6 turn out to be sharp (see, for example, [3, Examples 3.5 and 3.6]).

2.2 Proof of Theorem 2.5

The proof of Theorem 2.5 is quite long and relies on various simple properties of linear spaces in special position. The idea is to consider the images of the planes $\Lambda _1,\dots ,\Lambda _{d-1}$ under the projection $\pi _p\colon \mathbb {P}^n\dashrightarrow \mathbb {P}^{n-1}$ from a general point $p\in \Lambda _d$ . If $\pi _p(\Lambda _1),\dots ,\pi _p(\Lambda _{d-1})\subset \mathbb {P}^{n-1}$ is an indecomposable sequence of $(k-1)$ -planes satisfying $\operatorname {\mathrm {SP}}(n-k)$ , the assertion follows easily by induction. However, we need a big amount of work to prove the assertion when the sequence of linear spaces $\pi _p(\Lambda _i)$ is decomposable.

So, in order to prove Theorem 2.5, we firstly present various preliminary lemmas, which express properties of linear spaces in special position.

Lemma 2.9. Let $\Lambda _1,\dots ,\Lambda _d\subset \mathbb {P}^{n}$ be $(k-1)$ -planes that satisfy $\operatorname {\mathrm {SP}}(n-k)$ . If $\Lambda _1,\dots ,\Lambda _{d-1}$ are contained in a linear space S, then $\Lambda _d\subset S$ too.

Proof. See [3, Lemma 3.4].

Lemma 2.10. Let $\Lambda _1,\dots ,\Lambda _d\subset \mathbb {P}^n$ be $(k-1)$ -planes that satisfy $\operatorname {\mathrm {SP}}(n-k)$ . Let $1 \leq r \leq d-2$ be an integer and let $(p_{r+1},\dots ,p_{d-1})\in \Lambda _{r+1}\times \dots \times \Lambda _{d-1}$ be a $(d-1-r)$ -tuple such that

$$ \begin{align*}\operatorname{\mathrm{Span}}(p_{r+1},\dots,p_{d-1})\cap \Lambda_d = \emptyset.\end{align*} $$

Then $\Lambda _d\subset \operatorname {\mathrm {Span}}(\Lambda _1,\dots ,\Lambda _r ,p_{r+1},\dots ,p_{d-1})$ .

Proof. Let $S:=\operatorname {\mathrm {Span}}(\Lambda _1,\dots ,\Lambda _r ,p_{r+1},\dots ,p_{d-1})$ and let $s:=\dim S$ . If S is the whole projective space $\mathbb {P}^n$ , there is nothing to prove; hence, we assume hereafter $s < n$ . Moreover, $s \geq k - 1$ , and hence, $0 \leq s - k + 1 \leq n - k$ .

Let $P:=\operatorname {\mathrm {Span}}(p_{r+1},\dots ,p_{d-1})\subset S$ . By assumption, P does not meet $\Lambda _d$ ; therefore, $\dim P\leq n-k$ . Furthermore, $\dim P \leq s -k$ . Indeed, if the dimension of P were greater than $s - k$ , we would have $P \cap \Lambda _j\neq \emptyset $ for any $j=1,\dots ,r$ . Therefore, P would meet each $(k - 1)$ -plane $\Lambda _j$ , with $j=1,\dots ,d-1$ . Because of condition $\operatorname {\mathrm {SP}}(n-k)$ , any $(n-k)$ -plane $L \subset \mathbb {P}^n$ containing P should intersect also $\Lambda _d$ . Then we would have $P\cap \Lambda _d\neq \emptyset $ , a contradiction.

Now, let $T \subset S$ be a $(s -k + 1)$ -plane containing P. Then T meets each of the $(k-1)$ -planes $\Lambda _1,\dots ,\Lambda _{d-1}$ . By condition $\operatorname {\mathrm {SP}}(n-k)$ , any $(n-k)$ -plane L containing T must intersect the remaining plane $\Lambda _d$ . Therefore, $\Lambda _d\cap T\neq \emptyset $ for any $(s -k +1)$ -plane T as above. Finally, as $\Lambda _d \cap P =\emptyset $ by assumption, we deduce that $\Lambda _d\subset S$ .

Lemma 2.11. Let $\Lambda _1,\dots ,\Lambda _d\subset \mathbb {P}^n$ be $(k-1)$ -planes that satisfy $\operatorname {\mathrm {SP}}(n-k)$ . Let $2\leq r \leq d-1$ be an integer such that $\Lambda _{r+1},\dots ,\Lambda _d$ are not $\operatorname {\mathrm {SP}}(n-k)$ . Then

$$ \begin{align*}\dim\big(\operatorname{\mathrm{Span}}(\Lambda_1,\dots,\Lambda_r)\cap\operatorname{\mathrm{Span}}(\Lambda_{r+1},\dots,\Lambda_d)\big)\geq k-1.\\[-30pt]\end{align*} $$

Proof. By Lemma 2.9, the $(k-1)$ -plane $\Lambda _d$ is contained in $\operatorname {\mathrm {Span}}(\Lambda _1,\dots ,\Lambda _{d-1})$ . Hence, the assertion holds for $r=d-1$ .

So we assume $2\leq r \leq d-2$ . Since $\Lambda _{r+1},\dots ,\Lambda _d$ are not $\operatorname {\mathrm {SP}}(n-k)$ , there exists a $(n-k)$ -plane L intersecting all the $(k-1)$ -planes $\Lambda _{r+1},\dots ,\Lambda _d$ except one, say $\Lambda _d$ . For any $j=r+1,\dots ,d-1$ , we fix a point $p_j\in L\cap \Lambda _j$ . Let us define $P:=\operatorname {\mathrm {Span}}(p_{r+1},\dots ,p_{d-1})\subset L$ and $S:=\operatorname {\mathrm {Span}}(\Lambda _{1},\dots ,\Lambda _r,p_{r+1},\dots ,p_{d-1})$ . Hence, $P\cap \Lambda _d=\emptyset $ , and Lemma 2.10 ensures that $\Lambda _d\subset S$ .

Setting $\alpha :=\dim P$ and $\beta :=\dim \operatorname {\mathrm {Span}}(\Lambda _{1},\dots ,\Lambda _r)$ , we have that $\dim \operatorname {\mathrm {Span}}(P,\Lambda _d)=\alpha +k$ and $\dim S\leq \alpha +\beta +1$ . Since $\operatorname {\mathrm {Span}}(\Lambda _{1},\dots ,\Lambda _r)$ and $\operatorname {\mathrm {Span}}(P,\Lambda _d)$ are contained in S, we obtain

$$ \begin{align*}\dim \big(\operatorname{\mathrm{Span}}(\Lambda_{1},\dots,\Lambda_r)\cap \operatorname{\mathrm{Span}}(P,\Lambda_d)\big)\geq \beta+(\alpha+k)-(\alpha+\beta+1)=k-1.\end{align*} $$

Finally, as $\operatorname {\mathrm {Span}}(P,\Lambda _d)\subset \operatorname {\mathrm {Span}}(\Lambda _{r+1},\dots ,\Lambda _d)$ , the assertion follows.

Given a linear subspace $P\subset \mathbb {P}^n$ of dimension $\alpha :=\dim P\geq 0$ , we denote by $\pi _P\colon \mathbb {P}^n \dashrightarrow \mathbb {P}^{n-\alpha -1}$ the projection from P, which is defined on $\mathbb {P}^n\setminus P$ . Hence, if $\Lambda \subset \mathbb {P}^n$ is a linear subspace with $\dim (P\cap \Lambda )=\delta $ , then $\dim \pi _P(\Lambda )=\dim \Lambda -\delta -1$ , whereas for any linear subspace $R\subset \mathbb {P}^{n-\alpha -1}$ , the Zariski closure of $\pi _P^{-1}(R)\subset \mathbb {P}^n$ is a linear subspace of dimension $\dim R+\alpha +1$ which contains P.

Lemma 2.12. Let $\Lambda _1,\dots ,\Lambda _d\subset \mathbb {P}^n$ be $(k-1)$ -planes and let $P\subset \mathbb {P}^n$ be a linear subspace of dimension $\alpha \geq 0$ such that $P\cap \operatorname {\mathrm {Span}} \left(\Lambda _1,\dots ,\Lambda _d\right)=\emptyset $ . Using the notation above, consider the $(k-1)$ -planes $\Gamma _i:=\pi _P(\Lambda _i)\subset \mathbb {P}^{n-\alpha -1}$ , with $i=1,\dots ,d$ . Then $\Lambda _1,\dots ,\Lambda _d\subset \mathbb {P}^n$ are $\operatorname {\mathrm {SP}}(n-k)$ if and only if $\Gamma _1,\dots ,\Gamma _d$ are $\operatorname {\mathrm {SP}}(n-\alpha -1-k)$ .

Proof. Suppose that $\Lambda _1,\dots ,\Lambda _d\subset \mathbb {P}^n$ are $\operatorname {\mathrm {SP}}(n-k)$ and let $R\subset \mathbb {P}^{n-\alpha -1}$ be a $(n-\alpha -1-k)$ -plane intersecting $\Gamma _1,\dots ,\widehat {\Gamma _j},\ldots ,\Gamma _{d}$ , with $j\in \left\{1,\dots ,d\right\}$ . Then the linear subspace $L:=\overline {\pi _P^{-1}(R)}\subset \mathbb {P}^n$ is a $(n-k)$ -plane, which intersects $\Lambda _1,\dots ,\widehat {\Lambda _j},\ldots ,\Lambda _d$ . Hence, L must meet $\Lambda _j$ too, so that $R=\pi _P(L)$ intersects $\Gamma _j=\pi _P(\Lambda _j)$ . In conclusion, $\Gamma _1,\dots ,\Gamma _d$ are $\operatorname {\mathrm {SP}}(n-\alpha -1-k)$ .

Vice versa, suppose that $\Gamma _1,\dots ,\Gamma _d$ are $\operatorname {\mathrm {SP}}(n-\alpha -1-k)$ and let $L\subset \mathbb {P}^n$ be a $(n-k)$ -plane intersecting $\Lambda _1,\dots ,\widehat {\Lambda _j},\ldots ,\Lambda _d$ , with $j\in \left\{1,\dots ,d\right\}$ . For any $i=1,\dots ,\widehat {j},\ldots ,d$ , we fix a point $p_i\in \Lambda _i\cap L$ and we define $N:=\operatorname {\mathrm {Span}}(p_1,\dots ,\widehat {p_j},\ldots ,p_d)\subset L$ . Since $N\cap P=\emptyset $ , the linear space $\pi _P(N)\subset \mathbb {P}^{n-\alpha -1}$ has dimension $\dim N\leq \dim L=n-k$ , and it meets $\Gamma _1,\dots ,\widehat {\Gamma _j},\ldots ,\Gamma _{d}$ .

We claim that $\pi _P(N)$ must intersect $\Gamma _j$ too. Indeed, if $\dim \pi _P(N)\leq n-\alpha -1-k$ , condition $\operatorname {\mathrm {SP}}(n-\alpha -1-k)$ yields that any $(n-\alpha -1-k)$ -plane containing $\pi _P(N)$ must meet $\Gamma _j$ , so that $\pi _P(N)\cap \Gamma _j\neq \emptyset $ . If instead $\dim \pi _P(N)\geq n-\alpha -k$ , then $\pi _P(N)$ meets $\Gamma _j\subset \mathbb {P}^{n-\alpha -1}$ as $\dim \Gamma _j=k-1$ .

It follows that $\widetilde {N}:=\overline {\pi _P^{-1}\left(\pi _P(N)\right)}$ intersects $\Lambda _j$ . Denoting $S:= \operatorname {\mathrm {Span}} \left(\Lambda _1,\dots ,\Lambda _d\right)$ , we have $P\cap S=\emptyset $ by assumption, so the restriction $\pi _{P}|_S\colon S\longrightarrow \mathbb {P}^{n-\alpha -1}$ maps isomorphically to its image. Thus, $\widetilde {N}\cap S=N$ , as $\pi _P(\widetilde {N}\cap S)=\pi _P(N)$ . Since $\widetilde {N}\cap \Lambda _j\neq \emptyset $ and $\Lambda _j\subset S$ , we deduce that $N\subset L$ intersects $\Lambda _j$ , and we conclude that $\Lambda _1,\dots ,\Lambda _d\subset \mathbb {P}^n$ are $\operatorname {\mathrm {SP}}(n-k)$ .

Lemma 2.13. Let $\Lambda _1,\dots ,\Lambda _d\subset \mathbb {P}^n$ be $(k-1)$ -planes that satisfy $\operatorname {\mathrm {SP}}(n-k)$ and let $P\subset \mathbb {P}^n$ be a linear subspace of dimension $\alpha \geq 0$ . Let $2\leq r\leq d-1$ be an integer such that $P\cap \Lambda _i=\emptyset $ for any $i=1,\dots ,r$ , and $P\cap \Lambda _i\neq \emptyset $ for any $i=r+1,\dots ,d$ . Using the notation above, consider the $(k-1)$ -planes $\Gamma _i:=\pi _P(\Lambda _i)\subset \mathbb {P}^{n-\alpha -1}$ , with $i=1,\dots ,r$ . Then $\Gamma _1,\dots ,\Gamma _r$ are $\operatorname {\mathrm {SP}}(n-\alpha -1-k)$ .

Proof. By arguing as in the previous proof, we consider a $(n-\alpha -1-k)$ -plane $R\subset \mathbb {P}^{n-\alpha -1}$ intersecting $\Gamma _1,\dots ,\widehat {\Gamma _j},\ldots ,\Gamma _{r}$ , with $j\in \left\{1,\dots ,d\right\}$ . Then the linear subspace $L:=\overline {\pi _P^{-1}(R)}\subset \mathbb {P}^n$ is a $(n-k)$ -plane, which intersects $\Lambda _1,\dots ,\widehat {\Lambda _j},\ldots ,\Lambda _r$ and meets $\Lambda _{r+1},\dots ,\Lambda _d$ at P. Hence, L must intersect $\Lambda _j$ too since $\Lambda _1,\dots ,\Lambda _d$ are $\operatorname {\mathrm {SP}}(n-k)$ . Therefore, $R=\pi _P(L)$ intersects $\Gamma _j=\pi _P(\Lambda _j)$ , and we conclude that $\Gamma _1,\dots ,\Gamma _r$ are $\operatorname {\mathrm {SP}}(n-\alpha -1-k)$ .

Lemma 2.14. Let $\Lambda _1,\dots ,\Lambda _d\subset \mathbb {P}^n$ be $(k-1)$ -planes that satisfy $\operatorname {\mathrm {SP}}(n-k)$ . Let $2\leq r\leq \min \{d-2,n-k+1\}$ be an integer such that if $p\in \Lambda _{d}$ is a general point, then $p\not \in \Lambda _i$ for $i=1,\dots ,r$ and the $(k-1)$ -planes $\Gamma _1,\dots ,\Gamma _{r}\subset \mathbb {P}^n$ are $\operatorname {\mathrm {SP}}(n-k-1)$ , where $\pi _p\colon \mathbb {P}^n\dashrightarrow \mathbb {P}^{n-1}$ is the projection from p and $\Gamma _i:=\pi _p(\Lambda _i)$ for $i=1,\dots ,r$ . Then one of the following holds:

1. (A) $\Lambda _{d}\subset \operatorname {\mathrm {Span}}(\Lambda _1,\dots ,\Lambda _{r})$ ;

2. (B) $\Lambda _1,\dots ,\Lambda _{r}$ are $\operatorname {\mathrm {SP}}(n-k)$ .

Proof. Let $S:=\operatorname {\mathrm {Span}}(\Lambda _1,\dots ,\Lambda _{r})$ . If condition (A) does not hold, then $S\cap \Lambda _{d}$ is a (possibly empty) proper subset of $\Lambda _{d}$ . Suppose that L is a $(n-k)$ -plane intersecting $\Lambda _1,\dots ,\widehat {\Lambda _j},\ldots ,\Lambda _{r}$ . For any $i=1,\dots ,\widehat {j},\dots ,r$ , let us fix a point $p_i\in L\cap \Lambda _i$ and set $N:=\operatorname {\mathrm {Span}}\left(p_1,\dots ,\widehat {p_j},\dots ,p_r\right)\subset L\cap S$ .

Notice that a general point $p\in \Lambda _{d}$ is such that $p\not \in S$ , and consider $\pi _p(N)\subset \mathbb {P}^{n-1}$ . Then $\dim \pi _p(N)=\dim N\leq r-2 \leq n-k-1$ . By definition of N, the subspace $\pi _p(N)$ intersects $\Gamma _1,\dots ,\widehat {\Gamma _j},\ldots ,\Gamma _{r}$ and by condition $\operatorname {\mathrm {SP}}(n-k-1)$ , $\pi _p(N)$ must intersect $\Gamma _j$ too. Therefore, the linear subspace $\widetilde {N}:=\overline {\pi _p^{-1}\left(\pi _p(N)\right)}=\operatorname {\mathrm {Span}}(N,p)\subset \mathbb {P}^n$ intersects $\Lambda _j$ , and $\dim \widetilde {N}=\dim N+1\leq n-k$ . Moreover, since $\Lambda _j\subset S$ and $N=\widetilde {N}\cap S$ , we deduce that $N\cap \Lambda _j\neq \emptyset $ . Thus, L meets $\Lambda _j$ , and we conclude that $\Lambda _1,\dots ,\Lambda _{r}$ are $\operatorname {\mathrm {SP}}(n-k)$ , as in condition $(B)$ .

Lemma 2.15. Let $\Lambda _1,\dots ,\Lambda _d\subset \mathbb {P}^n$ be $(k-1)$ -planes $\operatorname {\mathrm {SP}}(n-k)$ . Consider a partition of the set $\{1,\dots ,d\}$ consisting of $m\geq 2$ parts of the form

(2.1) $$ \begin{align} D_1:=\{1,\dots,d_1\}, D_2:=\{d_1+1,\dots,d_1+d_2\},\dots,D_m:=\{\sum_{i=1}^{m-1} d_i+1,\dots,d\}, \end{align} $$

and for any $j=1,\dots , m$ , let $d_j$ be the cardinality of $D_j$ , and let $S_j$ denote the linear span of the $(k-1)$ -planes indexed by the part $D_j$ . Suppose that for any $j=1,\dots , m-1$ ,

1. (i) the $(k-1)$ -planes indexed by $D_j$ are $\operatorname {\mathrm {SP}}(n-k)$ ;

2. (ii) the $(k-1)$ -planes indexed by $\{1,\dots ,d\}\setminus D_j$ are not $\operatorname {\mathrm {SP}}(n-k)$ ;

3. (iii) $\dim S_j\leq d_j+k-3-\varepsilon _j$ for some $\varepsilon _j\geq 0$ .

Then

(2.2) $$ \begin{align} \dim\operatorname{\mathrm{Span}}\left(\Lambda_1,\dots,\Lambda_{d}\right)\leq\dim S_m+\sum_{j=1}^{m-1} \left(d_j-\varepsilon_j\right) -m. \end{align} $$

Proof. Consider the union J of $D_m$ and r other parts of (2.1), with $0\leq r\leq m-2$ . Then the $(k-1)$ -planes indexed by J are not $\operatorname {\mathrm {SP}}(n-k)$ ; otherwise, by assumption (i) and Remark 2.3, the union of J and $m-2-r$ parts not contained in J would index a collection of $(k-1)$ -planes $\operatorname {\mathrm {SP}}(n-k)$ , contradicting assumption (ii). In particular, the $(k-1)$ -planes indexed by $D_m$ are not $\operatorname {\mathrm {SP}}(n-k)$ .

In order to prove (2.2), we argue by induction on $m\geq 2$ . If $m=2$ , partition (2.1) is given by $D_1=\{1,\dots ,d_1\}$ and $D_2=\{d_1+1,\dots ,d\}$ . We note that $\Lambda _{1},\dots , \Lambda _{d_1}$ are $\operatorname {\mathrm {SP}}(n-k)$ and, in particular, $d_1\geq 2$ . Moreover, $\Lambda _{d_1+1},\dots , \Lambda _{d}$ are not $\operatorname {\mathrm {SP}}(n-k)$ , so Lemma 2.11 ensures that $\dim (S_1\cap S_2)\geq k-1$ . Since $\dim S_1\leq d_1+k-3-\varepsilon _1$ , we conclude that

$$ \begin{align*}\dim\operatorname{\mathrm{Span}}\left(\Lambda_1,\dots,\Lambda_{d}\right)=\dim \operatorname{\mathrm{Span}}(S_1, S_2) \leq \dim S_1 + \dim S_2 - (k-1) \leq \dim S_2 + d_1-\varepsilon_1-2, \end{align*} $$

as in (2.2).

Then we assume $m\geq 3$ . Let $P\subset S_{1}$ be a general subspace of dimension $\alpha :=\dim S_{1}-k+1$ . We distinguish three cases: 1) $P\cap S_m\neq \emptyset $ ; 2) $P\cap S_j\neq \emptyset $ for some $j=2,\dots ,m-1$ and $P\cap S_m=\emptyset $ ; 3) $P\cap S_j=\emptyset $ for any $j=2,\dots ,m$ . The key point in each case is showing that there exists a $S_j$ (or a linear space $\Sigma $ containing some spaces among $S_2,\dots ,S_m$ ) which intersects $S_1$ along a linear space of large dimension, so that $\dim \operatorname {\mathrm {Span}}\left(S_1,S_{j}\right)$ (resp. $\dim \operatorname {\mathrm {Span}}\left(S_1,\Sigma \right)$ ) is not too large.

Case 1. Suppose that $P\cap S_m\neq \emptyset $ . It follows by the generality of P that $\dim \left(S_{1}\cap S_m\right)\geq k-1$ , and hence,

(2.3) $$ \begin{align} \dim \operatorname{\mathrm{Span}}(S_{1},S_m)=\dim S_{1} + \dim S_{m}- \dim (S_{1}\cap S_m)\leq \dim S_m +d_{1}-\varepsilon_{1}-2. \end{align} $$

Then we consider the partition of $\{1,\dots ,d\}$ given by

(2.4) $$ \begin{align} D_2, D_3, \dots, D_{m-1},D_{1}\cup D_{m}, \end{align} $$

which is obtained by joining the first and the last part of (2.1). We note that (2.4) consists of $m-1$ parts satisfying assumptions (i), (ii) and (iii), and the span of the $(k-1)$ -planes indexed by the last part is $\operatorname {\mathrm {Span}}(S_{1},S_m)$ . Thus, we deduce by induction and (2.3) that

$$ \begin{align*} \dim\operatorname{\mathrm{Span}}(\Lambda_1,\dots,\Lambda_d)&\leq \dim\operatorname{\mathrm{Span}}(S_{1},S_m)+\sum_{j=2}^{m-1} (d_j - \varepsilon_j ) -(m-1)\\ &\leq \dim S_m+\sum_{j=1}^{m-1} (d_j-\varepsilon_j) -m-1, \end{align*} $$

so that (2.2) holds.

Case 2. Analogously, suppose that $P\cap S_j\neq \emptyset $ for some $j=2,\dots ,m-1$ and, without loss of generality, set $j=2$ . By arguing as above, the generality of P yields $\dim \left(S_{1}\cap S_{2}\right)\geq k-1$ and

(2.5) $$ \begin{align} \begin{aligned} \dim \operatorname{\mathrm{Span}}(S_{1},S_{2})&\leq \left(d_{1}+k-3-\varepsilon_{1}\right)+\left(d_{2}+k-3-\varepsilon_{2}\right)-(k-1)\\ &=d_{1}+d_{2}+k-3-\widetilde{\varepsilon}, \end{aligned} \end{align} $$

where $\widetilde {\varepsilon }:=\varepsilon _{1}+\varepsilon _{2}+2\geq 2$ . Then we consider the partition of $\{1,\dots ,d\}$ given by

(2.6) $$ \begin{align} D_1\cup D_2, D_3, \dots, D_{m}, \end{align} $$

which is obtained by joining the first and the second part of (2.1). Then (2.6) consists of $m-1$ parts, where the first part indexes a collection of $d_{1}+d_{2}$ linear spaces of dimension $(k-1)$ , which are $\operatorname {\mathrm {SP}}(n-k)$ by Remark 2.3, and their linear span is $\operatorname {\mathrm {Span}}(S_{1},S_{2})$ . In light of (2.5), assumptions (i), (ii) and (iii) are satisfied, and we deduce by induction that

$$ \begin{align*} \dim\operatorname{\mathrm{Span}}(\Lambda_1,\dots,\Lambda_d)&\leq \dim S_m+(d_{1}+d_{2}-\widetilde{\varepsilon})+\sum_{j=3}^{m-1} (d_j - \varepsilon_j ) -(m-1)\\ &\leq \dim S_m+\sum_{j=1}^{m-1} (d_j-\varepsilon_j) -m-1, \end{align*} $$

so that (2.2) holds.

Case 3. Finally, it remains to prove (2.2) when the $\alpha $ -plane $P\subset S_1$ is such that $P\cap S_j=\emptyset $ for any $j=2,\ldots ,m$ , where $\alpha :=\dim S_1-k+1$ . Since P is a subspace of codimension $k-1$ in $S_1$ , we deduce that $P\cap \Lambda _i=\emptyset $ if and only if $i=d_1+1,\dots ,d$ . Then we consider the projection $\pi _P\colon \mathbb {P}^n \dashrightarrow \mathbb {P}^{n-\alpha -1}$ from P, and we set $\Gamma _i:=\pi _P(\Lambda _i)$ for any $i=d_1+1,\dots ,d$ . Denoting $S^{\prime }_j:=\pi _P(S_j)$ for any $j=2,\ldots ,m$ , it is easy to see that each $S^{\prime }_j$ is spanned by the $(k-1)$ -planes $\Gamma _i$ indexed by the part $D_j$ . We would like to use induction on $\Gamma _{d_1+1},\dots ,\Gamma _d\subset \mathbb {P}^{n-\alpha -1}$ with respect to the partition of $\{d_1+1,\dots ,d\}$ given by

(2.7) $$ \begin{align} D_2, D_3,\dots, D_m, \end{align} $$

which is obtained by removing the first part of (2.1). Thanks to Lemma 2.13, $\Gamma _{d_1+1},\dots ,\Gamma _d\subset \mathbb {P}^{n-\alpha -1}$ are $\operatorname {\mathrm {SP}}(n-\alpha -1-k)$ , and the $(k-1)$ -planes $\Gamma _{i}$ indexed by $D_j$ satisfy assumption (i) for any $j=2,\dots ,m-1$ . Moreover, $\dim S^{\prime }_j=\dim S_j$ for any $j=2,\ldots ,m$ , so that assumption (iii) holds.

Case 3(a). Suppose in addition that $\Gamma _{d_1+1},\dots ,\Gamma _d\subset \mathbb {P}^{n-\alpha -1}$ satisfy assumption (ii) with respect to partition (2.7). We deduce by induction that

$$ \begin{align*}\dim\operatorname{\mathrm{Span}}(\Gamma_{d_1+1},\dots,\Gamma_d)\leq \dim S_m+\sum_{j=2}^{m-1} (d_j - \varepsilon_j ) -(m-1). \end{align*} $$

Let $\Sigma :=\operatorname {\mathrm {Span}}\left(\Lambda _{d_1+1},\dots ,\Lambda _d\right)$ . In light of assumptions (i) and (ii), Lemma 2.11 ensures that $\dim (S_1\cap \Sigma )=k-1+\gamma $ for some $\gamma \geq 0$ . As P is a general linear space of codimension $k-1$ in $S_1$ , it follows that $\dim (P\cap \Sigma )=\gamma $ . Moreover, $\pi _P(\Sigma )=\operatorname {\mathrm {Span}}(\Gamma _{d_1+1},\dots ,\Gamma _d)$ , and hence,

$$ \begin{align*}\dim \Sigma=\dim\operatorname{\mathrm{Span}}(\Gamma_{d_1+1},\dots,\Gamma_d) +\gamma +1\leq \dim S_m+\sum_{j=2}^{m-1} (d_j - \varepsilon_j ) -m+2+\gamma. \end{align*} $$

Therefore, $\dim \operatorname {\mathrm {Span}}\left(\Lambda _{1},\dots ,\Lambda _d\right)=\dim \operatorname {\mathrm {Span}}(S_1,\Sigma )=\dim S_1+\dim \Sigma -\dim \left(S_1\cap \Sigma \right)$ , so that

$$ \begin{align*} \dim \operatorname{\mathrm{Span}}\left(\Lambda_{1},\dots,\Lambda_d\right)&\leq (d_1+k-3-\varepsilon_1)+ \left(\dim S_m+\sum_{j=2}^{m-1} (d_j - \varepsilon_j ) -m+2+\gamma\right) -(k-1+\gamma)\\ &=\dim S_m+\sum_{j=1}^{m-1} (d_j - \varepsilon_j )-m, \end{align*} $$

which is (2.2).

Case 3(b). Finally, suppose that $\Gamma _{d_1+1},\dots ,\Gamma _d$ do not satisfy assumption (ii) with respect to (2.7). Without loss of generality, we assume that $\Gamma _{d_1+d_2+1},\dots ,\Gamma _d$ are $\operatorname {\mathrm {SP}}(n-\alpha -1-k)$ (i.e., assumption (ii) fails for the $(k-1)$ -planes $\Gamma _{i}$ such that $i\in \{d_1+1,\dots ,d\}\setminus D_2$ ).

By Lemma 2.12, the $(k-1)$ -planes $\Gamma _{i}\subset \mathbb {P}^{n-\alpha -1}$ indexed by $D_m$ are not $\operatorname {\mathrm {SP}}(n-\alpha -1-k)$ because at the beginning of the proof, we pointed out that the corresponding $\Lambda _{i}\subset \mathbb {P}^{n}$ are not $\operatorname {\mathrm {SP}}(n-k)$ . Let $1\leq t\leq m-3$ be an integer such that the $(k-1)$ -planes $\Gamma _{i}$ indexed by the union of $D_m$ and t other parts of (2.7) satisfy the hypothesis of the lemma. The existence of t is granted by the fact that the $(k-1)$ -planes $\Gamma _{i}$ such that $i\in D_m$ are not $\operatorname {\mathrm {SP}}(n-\alpha -1-k)$ , whereas $\Gamma _{d_1+d_2+1},\dots ,\Gamma _d$ are $\operatorname {\mathrm {SP}}(n-\alpha -1-k)$ . Without loss of generality, we assume that the last $t+1$ parts of (2.7) – that is, $D_{m-t},\dots ,D_m$ – satisfy the hypotheses of the lemma.

Let $T:=\operatorname {\mathrm {Span}}\left(S_{m-t},\dots ,S_{m}\right)$ be the span of the $(k-1)$ -planes $\Lambda _{i}\subset \mathbb {P}^{n}$ with $i\in D_{m-t}\cup \dots \cup D_m$ . We note that $P\cap T\neq \emptyset $ ; otherwise, Lemma 2.12 would assure that these $(k-1)$ -planes are $\operatorname {\mathrm {SP}}(n-k)$ , contradicting the remark at the beginning of the proof. Let $\delta :=\dim (P\cap T)\geq 0$ , so that $\dim (S_1\cap T)=k-1+\delta $ . Since $\dim S_j'=\dim S_j$ for any $j=2,\dots , m$ , using induction on the $(k-1)$ -planes $\Gamma _{i}\subset \mathbb {P}^{n-1}$ indexed by $D_{m-t}\cup \dots \cup D_m$ , we obtain $\dim \pi _P(T)\leq \dim S_m+\sum _{j=m-t}^{m-1} (d_j- \varepsilon _j) -t-1$ . Then we set $\Sigma :=\overline {\pi ^{-1}_P\left(\pi _P(T)\right)}$ , and we deduce that

(2.8) $$ \begin{align} \dim \Sigma= \dim \pi_P(T)+\dim P+1 \leq \dim S_m+\sum_{j=m-t}^{m-1} (d_j- \varepsilon_j) +(d_1-\varepsilon_1)-(t+2). \end{align} $$

We point out that $S_1\cap \Sigma $ contains $\operatorname {\mathrm {Span}}(P,S_1\cap T)$ , whose dimension is

$$ \begin{align*}\dim \operatorname{\mathrm{Span}}(P,S_1\cap T)=\dim P+\dim(S_1\cap T)-\dim(P\cap T) =\dim P+k-1+\delta-\delta=\dim S_1. \end{align*} $$

Therefore, $S_1\subset \Sigma $ , so that $\Sigma =\operatorname {\mathrm {Span}}\left(S_1,S_{m-t},\dots ,S_m\right)$ .

To conclude, we consider the partition of $\{1,\dots ,d\}$ given by

(2.9) $$ \begin{align} D_2, D_3, \dots,D_{m-t-1}, D_1\cup D_{m-t}\cup\dots\cup D_m, \end{align} $$

which is obtained from (2.1) by joining the first part and the last $t+1$ parts. By assumptions, the $(k-1)$ -planes $\Lambda _1,\dots ,\Lambda _d$ satisfy the hypotheses of the lemma with respect to partition (2.9). So we conclude by induction and inequality (2.8) that

$$ \begin{align*} \dim\operatorname{\mathrm{Span}}(\Lambda_1,\dots,\Lambda_d)&\leq \dim \operatorname{\mathrm{Span}}\left(S_1,S_{m-t},\dots,S_m\right)+\sum_{j=2}^{m-t-1} (d_j - \varepsilon_j ) -(m-t-1)\\ &\leq \dim S_m+\sum_{j=1}^{m-1} (d_j-\varepsilon_j) -m-1, \end{align*} $$

which implies (2.2).

We can finally prove Theorem 2.5.

Proof of Theorem 2.5.

If $n\leq d+k-3$ , the assertion is trivial. So we assume hereafter that $n\geq d+k-2$ . We argue by induction on $d\geq 2$ .

If $d=2$ , the $(k-1)$ -planes $\Lambda _1,\Lambda _2\subset \mathbb {P}^n$ are $\operatorname {\mathrm {SP}}(n-k)$ if and only if they coincide, so the statement is true.

Then we assume $d\geq 3$ . We point out that there exists $j\in \{1,\dots ,d\}$ such that $\Lambda _j\neq \Lambda _i$ for all $i\neq j$ ; otherwise, by grouping the indices of coincident $(k-1)$ -planes, we would obtain a partition of $\{1,\dots ,d\}$ , where each part consists of at least 2 coincident $(k-1)$ -planes, which are trivially $\operatorname {\mathrm {SP}}(n-k)$ , but this is impossible as the sequence $\Lambda _1,\dots ,\Lambda _d$ is indecomposable. Without loss of generality, we assume $j=d$ .

Therefore, given a general point $p\in \Lambda _d$ , we have that $p\not \in \Lambda _i$ for any $i=1,\dots ,d-1$ . Let us consider the projection $\pi _p:\mathbb {P}^n\dashrightarrow \mathbb {P}^{n-1}$ from p, and for any $i=1,\dots ,d-1$ , let us set $\Gamma _i:=\pi _p(\Lambda _i)$ . By Lemma 2.13, the $(k-1)$ -planes $\Gamma _1,\dots ,\Gamma _{d-1}\subset \mathbb {P}^{n-1}$ are $\operatorname {\mathrm {SP}}(n-k-1)$ .

If the sequence $\Gamma _1,\dots ,\Gamma _{d-1}$ is indecomposable, we deduce by induction that $\dim \operatorname {\mathrm {Span}}\left(\Gamma _1,\dots ,\Gamma _{d-1}\right)\leq d+k-4$ . Therefore, the closure of $\pi _p^{-1}\left(\operatorname {\mathrm {Span}}\left(\Gamma _1,\dots ,\Gamma _{d-1}\right)\right)$ is a linear space $\Sigma \subset \mathbb {P}^n$ of dimension at most $d+k-3$ . Since $\Lambda _1,\dots ,\Lambda _{d-1}\subset \Sigma $ , Lemma 2.9 implies that $\Lambda _{d}\subset \Sigma $ too. Thus, we conclude that $\dim \operatorname {\mathrm {Span}}\left(\Lambda _1,\dots ,\Lambda _{d}\right)\leq d+k-3$ , as wanted.

So we assume that $\Gamma _1,\dots ,\Gamma _{d-1}$ is a decomposable sequence (i.e., there exists a partition of $\{1,\dots ,d-1\}$ in $m\geq 2$ parts, where each part $\{i_1,\dots ,i_t\}$ is such that the corresponding $(k-1)$ -planes $\Gamma _{i_1},\dots ,\Gamma _{i_t}$ are $\operatorname {\mathrm {SP}}(n-k-1)$ ). Let

(2.10) $$ \begin{align} D_1:=\{1,\dots,d_1\}, D_2:=\{d_1+1,\dots,d_1+d_2\},\dots, D_m:=\{\sum_{i=1}^{m-1} d_i+1,\dots,d-1\} \end{align} $$

denote such a partition, where the part $D_j$ has cardinality $d_j$ and $\sum _{j=1}^md_j=d-1$ . Furthermore, up to taking a finer partition, we may assume that each part indexes an indecomposable sequence of $(k-1)$ -planes $\Gamma _i$ . We point out that the partition does not depend on the choice of the general point $p\in \Lambda _d$ because $\Lambda _d$ is irreducible and the number of partitions of $\{1,\dots ,d-1\}$ is finite. For any $j=1,\dots , m$ , let

$$ \begin{align*}S^{\prime}_j:=\operatorname{\mathrm{Span}}\left(\Gamma_i\left|i\in D_j\right.\right)\subset \mathbb{P}^{n-1} \quad\text{and}\quad S_j:=\operatorname{\mathrm{Span}}\left(\Lambda_i\left|i\in D_j\right.\right)\subset \mathbb{P}^{n}. \end{align*} $$

By induction, $\dim S^{\prime }_j\leq d_j+k-3$ for any $j=1,\dots ,m$ .

Since we assumed $n\geq d+k-2$ , we have that $2\leq d_j<d-1\leq n-k+1$ for any $j=1,\dots ,m$ . Hence, Lemma 2.14 ensures that for any $j=1,\dots ,m$ , either $\Lambda _d\subset S_j$ – as in case (A) – or the $(k-1)$ -planes $\Lambda _i$ indexed by $D_j$ are $\operatorname {\mathrm {SP}}(n-k)$ , as in case (B). Let $0\leq t\leq m$ be the largest number of parts such that $\Lambda _d\subset S_j$ and, without affecting generality, assume that this happens for $j=m-t+1,\dots ,m$ . Therefore,

(2.11) $$ \begin{align} \dim\operatorname{\mathrm{Span}}(S_{m-t+1},\dots,S_m)\leq \sum_{j=m-t+1}^m(d_j+k-3)-(t-1)(k-1)=\sum_{j=m-t+1}^m d_j -2t+k-1. \end{align} $$

If $t=m$ – that is, $\Lambda _d\subset S_j$ for any $j=1,\dots ,m$ – the previous formula gives

$$ \begin{align*}\dim\operatorname{\mathrm{Span}}(\Lambda_{1},\dots,\Lambda_d)=\dim\operatorname{\mathrm{Span}}(S_{1},\dots,S_m)\leq \sum_{j=1}^m d_j -2m+k-1, \end{align*} $$

so the assertion follows as $ \sum _{j=1}^m d_j=d-1$ and $m\geq 2$ .

If instead $0\leq t<m$ , the first $m-t$ parts of (2.10) satisfy condition (B) of Lemma 2.14, and they do not satisfy condition (A) (i.e., for any $j=1,\dots ,m-t$ , the $(k-1)$ -planes $\Lambda _i$ indexed by $D_j$ are $\operatorname {\mathrm {SP}}(n-k)$ and $\Lambda _d\not \subset S_j$ ). In particular, $p\not \in S_j$ because $p\in \Lambda _d$ is a general point. Therefore, the $(k-1)$ -planes indexed by $D_j$ give an indecomposable sequence; otherwise, by Lemma 2.12, the corresponding sequence of $\Gamma _i=\pi _p(\Lambda _i)$ would be decomposable, but this is not the case. Then we deduce by induction that $\dim S_j\leq d_j+k-3$ for any $j=1,\ldots , m-t$ .

Finally, let us consider the partition of $\{1,\dots ,d\}$ in $m-t+1$ parts given by

(2.12) $$ \begin{align} D_1, D_2,\dots,D_{m-t}, D_{m-t+1}\cup \dots\cup D_m\cup\{d\}, \end{align} $$

where the first $m-t$ parts are those of (2.10), whereas the last part is the union of $\{d\}$ with the last t parts of (2.10). By summing up, the sequence $\Lambda _1,\dots ,\Lambda _d$ is indecomposable, and for any $j=1,\dots ,m-t$ , the part $D_j$ indexes a sequence of $(k-1)$ -planes which are $\operatorname {\mathrm {SP}}(n-k)$ , whose span is $S_j$ and satisfies $\dim S_j\leq d_j+k-3$ . Therefore, the assumptions of Lemma 2.15 hold with respect to partition (2.12).

Concerning the last part of (2.12), let B be the span of the corresponding $(k-1)$ -planes. If $t=0$ , the last part is just $\{d\}$ and $\dim B=\dim \Lambda _d =k-1$ . If instead $t>0$ , then $B=\operatorname {\mathrm {Span}}(S_{m-t+1},\dots ,S_m)$ because $\Lambda _d\subset S_j$ for any $j=m-t+1,\dots ,d$ , and $\dim B$ is bounded by (2.11). Thus, for any $t\geq 0$ , we obtain from Lemma 2.15 that

$$ \begin{align*}\dim\operatorname{\mathrm{Span}}(\Lambda_1,\dots,\Lambda_d)\leq \dim B+ \sum_{j=1}^{m-t}d_j-(m-t+1) \leq \sum_{j=1}^m d_j +k-m-t-2. \end{align*} $$

In particular, the assertion holds since $ \sum _{j=1}^m d_j=d-1$ , $m\geq 2$ and $t\geq 0$ .

2.3 Further remarks

To conclude this section, we collect some remarks and possible developments concerning linear subspaces of $\mathbb {P}^n$ in special position and, more generally, points of $\mathbb {G}(k-1,n)$ satisfying the Cayley–Bacharach condition with respect to $\left|\mathcal {O}_{\mathbb {G}(k-1,n)}(r)\right|$ , for some $r\geq 1$ .

Remark 2.16. In this section, we discussed the dimension of the span of $(k-1)$ -dimensional linear spaces in $\mathbb {P}^n$ as being $\operatorname {\mathrm {SP}}(n-k)$ is a property descending from the Cayley–Bacharach condition with respect to $\left|\mathcal {O}_{\mathbb {G}(k-1,n)}(1)\right|$ .

More generally, it would be interesting to investigate the same problem for $(k-1)$ -dimensional linear spaces in $\mathbb {P}^n$ parameterized by a set $\Gamma \subset \mathbb {G}(k-1,n)$ of points, which satisfies the Cayley–Bacharach condition with respect to $\left|\mathcal {O}_{\mathbb {G}(k-1,n)}(r)\right|$ for larger values of $r\geq 1$ . Moreover, this may lead to some progress in the study of measures of irrationality of Fano schemes of complete intersections.

We recall that the Fano scheme of a projective variety $X\subset \mathbb {P}^n$ is the scheme $F_k(X)\subset \mathbb {G}(k,n)$ parameterizing k-dimensional linear spaces contained in X. In [15], the authors study the geometry of lines in special position in $\mathbb {P}^4$ in order to determine the covering gonality, the connecting gonality and the degree of irrationality of the Fano surface $F_1(X)$ of a smooth cubic threefold $X\subset \mathbb {P}^4$ . In this setting, the connection between measures of irrationality and lines in special position depends on the following fact. By the same techniques we use in the next section, the points of $F_1(X)$ computing its covering gonality (or another measure of irrationality, as, for example, the connecting gonality and the degree of irrationality) satisfy the Cayley–Bacharach condition with respect to the canonical linear series of $F_1(X)$ , which is the linear series $\left|\mathcal {O}_{F_1(X)}(1)\right|$ cut out on $F_1(X)$ by hyperplanes of $\mathbb {P}^{9}$ under the Plücker embedding of $\mathbb {G}(1,4)$ .

The same techniques apply in general to Fano schemes $F_k(X)\subset \mathbb {G}(k,n)$ of smooth complete intersections $X\subset \mathbb {P}^n$ of large degree, whose canonical bundle has the form $\mathcal {O}_{F_k(X)}(r)$ , with $r\geq 1$ (see, for example, [12, Remarques 3.2]), hence motivating the study of the geometry of k-planes satisfying the Cayley–Bacharach condition with respect to $\left|\mathcal {O}_{\mathbb {G}(k,n)}(r)\right|$ .

Remark 2.17. From [17], we recall the following definition: we say that a union $\mathcal {P}=P_1\cup \dots \cup P_m\subset \mathbb {P}^n$ of positive-dimensional linear spaces is a plane configuration of length m and dimension $\dim \mathcal {P}:=\sum _{i=1}^m\dim P_i$ . Given two positive integers d and r, the authors study the least dimension of a plane configuration containing a given set $\Gamma \subset \mathbb {P}^n$ of d points satisfying the Cayley–Bacharach condition with respect to $|\mathcal {O}_{\mathbb {P}^n}(r)|$ .

We note that Corollary 2.6 can be rephrased in terms of plane configurations. Namely, let $\Lambda _1,\dots ,\Lambda _d\subset \mathbb {P}^{n}$ be linear spaces of dimension $(k-1)$ as in Corollary 2.6. By arguing as in its proof, we easily see that $\Lambda _1,\dots ,\Lambda _d$ are contained in a plane configuration $\mathcal {P}=P_1\cup \dots \cup P_m\subset \mathbb {P}^n$ of dimension $\dim \mathcal {P}=d+ m(k-3)$ , where each $P_i$ is the linear span of the $(k-1)$ -planes indexed by the i-th part.

So, in analogy with [17], it would be interesting to determine the least dimension of a plane configuration containing the $(k-1)$ -planes $\Lambda _1,\dots ,\Lambda _d\subset \mathbb {P}^{n}$ parameterized by a set $\Gamma \subset \mathbb {G}(k-1,n)$ of d points, which satisfies the Cayley–Bacharach condition with respect to $\left|\mathcal {O}_{\mathbb {G}(k-1,n)}(r)\right|$ for some $r\geq 1$ .

Remark 2.18. It follows from [18, Lemma 2.5] and [22, Theorem A] that if $\Gamma \subset \mathbb {P}^{n}$ is a finite set of points satisfying the Cayley–Bacharach condition with respect to $|\mathcal {O}_{\mathbb {P}^n}(r)|$ and $|\Gamma |\leq h(r-3+h)-1$ for some $r\geq 1$ and $2\leq h\leq 5$ , then $\Gamma $ lies on a curve of degree $h-1$ (actually, this is true for all $h\geq 2$ when $n=2$ ). Moreover, questions about analogous results are discussed in [17, Section 7.3].

Similarly, given a finite set $\Gamma \subset \mathbb {G}(k-1,n)$ of points satisfying the Cayley–Bacharach condition with respect to $\left|\mathcal {O}_{\mathbb {G}(k-1,n)}(r)\right|$ and having small cardinality, it would be interesting to understand whether the $(k-1)$ -planes parameterized by $\Gamma $ lie on a k-dimensional variety of low degree. For instance, looking at [3, Examples 3.5 and 3.6] and [15, Proposition 5.2], one deduces that $3$ lines satisfying $\operatorname {\mathrm {SP}}(n-2)$ lie on a plane and $4$ lines satisfying $\operatorname {\mathrm {SP}}(n-2)$ lie on a – possibly reducible – quadric surface.

3 Covering gonality of $C^{(3)}$ and $C^{(4)}$

In this section, we study the covering gonality of symmetric products of curves, and, in particular, we aim to prove Theorem 1.1.

3.1 Preliminaries

To start, we recall some preliminary facts concerning covering gonality of projective varieties and linear series on smooth curves.

Let X be an irreducible complex projective variety of dimension n.

Definition 3.1. A covering family of d-gonal curves on X consists of a smooth family $\mathcal {E}\stackrel {\pi }{\longrightarrow } T$ of irreducible curves (where the base variety T is allowed to be quasi-projective), endowed with a dominant morphism $f\colon \mathcal {E}\longrightarrow X$ such that for general $t\in T$ , the fiber $E_t:=\pi ^{-1}(t)$ is a smooth curve with gonality $\operatorname {\mathrm {gon}}(E_t)=d$ and the restriction $f_{t}\colon E_t\longrightarrow X$ of f is birational onto its image.

It is worth noticing that the covering gonality of X coincides with the least integer $d>0$ such that a covering family of d-gonal curves exists (see, for example, [15, Lemma 2.1]).

Given a covering family $\mathcal {E}\stackrel {\pi }{\longrightarrow } T$ of d-gonal curves, both the varieties T and $\mathcal {E}$ can be assumed to be smooth, with $\dim (T)=n-1$ (see [5, Remark 1.5]). Moreover, up to base changing T, there is a commutative diagram

(3.1)

where the restriction $\varphi _t\colon E_t \longrightarrow \{t\}\times \mathbb {P}^1 \cong \mathbb {P}^1$ is a d-gonal map (cf. [3, Example 4.7]).

Turning to symmetric products of curves, let C be a smooth projective curve of genus $g\geq 2$ and let $C^{(k)}$ be its k-fold symmetric product, with $2\leq k\leq g-1$ . Let ${\phi \colon C\longrightarrow \mathbb {P}^{g-1}}$ be the canonical map and let $\mathbb {G}(k-1,g-1)$ denote the Grassmann variety of ${(k-1)}$ -planes in $\mathbb {P}^{g-1}$ . Thanks to the General Position Theorem (see [1, p. 109]), we can define the Gauss map

$$ \begin{align*}\gamma\colon C^{(k)}\dashrightarrow \mathbb{G}(k-1,g-1),\end{align*} $$

which sends a general point ${p_1+\dots +p_k\in C^{(k)}}$ to the point of $\mathbb {G}(k-1,g-1)$ parameterizing the $(k-1)$ -plane $\operatorname {\mathrm {Span}}\left(\phi (p_1),\dots ,\phi (p_k)\right)\subset \mathbb {P}^{g-1}$ . We would also like to recall that $C^{(k)}$ is a variety of general type, where $H^0\left(C^{(k)},\omega _{C^{(k)}}\right)\cong \bigwedge ^k H^0\left(C,\omega _{C}\right)$ and the canonical map factors through the Gauss map and the Plücker embedding $p\colon \mathbb {G}(k-1,g-1)\longrightarrow \mathbb {P}^{{g \choose k }-1}$ – that is,

(3.2)

(cf. [19]).

The following theorem is the key result connecting covering families of irreducible d-gonal curves on $C^{(k)}$ and $(k-1)$ -planes of $\mathbb {P}^{g-1}$ in special position with respect to $(g-1-k)$ -planes. It is implicit in [3, Section 4], but we present a proof for the sake of completeness.

Theorem 3.2. Let C be a smooth projective curve of genus $g\geq 2$ and let $C^{(k)}$ be its k-fold symmetric product, with $2\leq k\leq g-1$ . Let $\mathcal {E}\stackrel {\pi }{\longrightarrow } T$ be a covering family of d-gonal curves on $C^{(k)}$ . Using notation as above, consider a general point $(t,y)\in T\times \mathbb {P}^1$ and let $\varphi ^{-1}(t,y)=\left\{x_1,\ldots ,x_d\right\}\subset E_t$ be its fiber. Then the $(k-1)$ -planes parameterized by $(\gamma \circ f) (x_1),\dots ,(\gamma \circ f) (x_d)\in \mathbb {G}(k-1,g-1)$ are in special position with respect to $(g-1-k)$ -planes of $\mathbb {P}^{g-1}$ .

Proof. According to [3, Definition 4.1] and [3, Example 4.7], the covering family $\mathcal {E}\stackrel {\pi }{\longrightarrow } T$ induces a correspondence with null trace on $(T\times \mathbb {P}^1)\times C^{(k)}$ given by

$$ \begin{align*}\Gamma:=\overline{\left\{\left.\big((t,y),P\big)\in (T\times \mathbb{P}^1)\times C^{(k)}\right|P\in f(E_t) \text{ and } f_t^{-1}(P)\in\varphi^{-1}(t,y) \right\}} \end{align*} $$

(i.e., the general point $\big ((t,y),P\big )\in \Gamma $ is such that $P\in f(E_t)$ , and the preimage of P on $E_t$ maps to $(t,y)$ under the d-gonal map $\varphi _t\colon E_t \longrightarrow \{t\}\times \mathbb {P}^1 \cong \mathbb {P}^1$ ). In particular, the projection $\pi _1\colon \Gamma \longrightarrow T\times \mathbb {P}^1$ has degree d since the general fiber $\pi _1^{-1}(t,y)$ consists of d points $\big ((t,y),P_i\big )$ , where $P_i:=f_t(x_i)$ and $\varphi ^{-1}(t,y)=\left\{x_1,\ldots ,x_d\right\}\subset E_t$ .

Therefore, [3, Proposition 4.2] ensures that for any $i=1,\dots ,d$ and for any canonical divisor $K\in \left|\omega _{C^{(k)}}\right|$ containing $P_1,\dots ,\widehat {P_i},\dots ,P_d$ , we have $P_i\in K$ . In addition, it follows from [3, Lemma 2.1] that any $(g-1-k)$ -plane $L\subset \mathbb {P}^{g-1}$ defines a canonical divisor $K_L$ on $C^{(k)}$ , and $P\in K_L$ if and only if $\gamma (P)\in \mathbb {G}(k-1,g-1)$ is a point of the Schubert cycle $\sigma _1(L):=\left\{\left.[\Lambda ]\in \mathbb {G}(k-1,g-1) \right|\Lambda \cap L\neq \emptyset \right\}$ , which parameterizes $(k-1)$ -planes of $ \mathbb {P}^{g-1}$ intersecting L. Thus, for any $i=1,\dots ,d$ and for any $(g-1-k)$ -plane $L\subset \mathbb {P}^{g-1}$ such that $(\gamma \circ f) (x_1),\dots ,\widehat {(\gamma \circ f) (x_i)},\dots ,(\gamma \circ f) (x_d)\in \sigma _1(L)$ , we have that $(\gamma \circ f) (x_i)\in \sigma _1(L)$ , i.e. the $(k-1)$ -planes parameterized by $(\gamma \circ f) (x_1),\dots ,(\gamma \circ f) (x_d)\in \mathbb {G}(k-1,g-1)$ are in special position with respect to $(g-1-k)$ -planes of $\mathbb {P}^{g-1}$ .

Remark 3.3. We note that for any smooth curve C of genus $g\geq 0$ , its k-fold symmetric product is covered by the family of curves $\mathcal {C}\stackrel {\pi }{\longrightarrow } C^{(k-1)}$ , where the fiber over $P=p_1+\dots +p_{k-1}\in C^{(k-1)}$ is the curve $C_P:=\left\{\left.p_1+\dots +p_{k-1}+q\in C^{(k)}\right|q\in C\right\}$ isomorphic to C. Thus,

(3.3) $$ \begin{align} \operatorname{\mathrm{cov.gon}}(C^{(k)})\leq \operatorname{\mathrm{gon}}(C). \end{align} $$

Given two positive integers r and d, let $W^r_d$ be the subvariety of $\operatorname {\mathrm {Pic}}^d(C)$ , which parameterizes complete linear series on C having degree d and dimension at least r (cf. [1, p. 153]). Moreover, we recall that a $\mathfrak {g}^r_d$ on C is a (possibly non-complete) linear series of degree d and dimension r. Finally, before proving Theorem 1.1, we point out two elementary facts involved in its proof.

Remark 3.4. Let $C\subset \mathbb {P}^{g-1}$ be the canonical model of a smooth non-hyperelliptic curve and let $D=q_1+\dots +q_s\in \operatorname {Div}(C)$ be a reduced divisor. It follows from the geometric version of the Riemann–Roch theorem (cf. [1, p. 12]) that the complete linear series $|D|$ is a $\mathfrak {g}^r_s$ with $r= s-1-\dim \operatorname {\mathrm {Span}}(q_1,\dots ,q_s)$ . If, in addition, $r\geq 1$ , then

(3.4) $$ \begin{align} \dim \operatorname{\mathrm{Span}}(q_1,\dots,q_s)\geq\operatorname{\mathrm{gon}}(C)-2. \end{align} $$

Indeed, $\dim |D-(r-1)p|\geq \dim |D|-(r-1)=1$ for any $p\in C$ , and hence, $\deg (D-(r-1)p)=s-r+1\geq \operatorname {\mathrm {gon}}(C)$ . Since $r= s-1-\dim \operatorname {\mathrm {Span}}(q_1,\dots ,q_s)$ , we obtain (3.4).

Lemma 3.5. Let C be a smooth non-hyperelliptic curve of genus $g\geq 3$ and, for some effective divisor $D\in \operatorname {Div}(C)$ , let $|D|$ be a $\mathfrak {g}^r_d$ such that $r\geq 2$ and $\dim |D-R|= 0$ for any $R\in C^{(r)}$ . Then $|D|$ is a very ample $\mathfrak {g}^{2}_{d}$ on C.

Proof. We note that $d\geq 2r\geq 4$ by Clifford’s theorem (see, for example, [1, p. 107]). For any $R\in C^{(r)}$ , we have that $\dim |D-R|= \dim |D|-r= 0$ . Since $\dim |D-p|\geq \dim |D|-1$ for any $p\in C$ , it follows that $\dim |D-p-q|=\dim |D|-2$ for any $p,q\in C$ , so that $|D|$ is very ample (cf. [21, Proposition V.4.20]).

By contradiction, suppose that $r\geq 3$ . For any $B\in C^{(b)}$ having $1\leq b\leq r-2$ , the divisor $D-B\in \operatorname {Div}(C)$ is such that $\dim |D-B|=r-b\geq 2$ and $\dim |D-B-A|=0$ for any $A\in C^{(r-b)}$ . Thus, we may argue as above, and we deduce that $|D-B|$ is a very ample $\mathfrak {g}^{r-b}_{d-b}$ . In particular, C admits a very ample $\mathfrak {g}^2_e$ with $e:=d-r+2\geq 4$ , so that $g=\frac {(e-1)(e-2)}{2}$ . Analogously, C is endowed with a very ample $\mathfrak {g}^3_{e+1}$ , and Castelnuovo’s bound (cf. [1, p. 107]) implies that $g\leq m(m-1)+m\varepsilon $ , where $m=\frac {e-\varepsilon }{2}$ and $\varepsilon \in \{0,1\}$ . Thus, we obtain

(3.5) $$ \begin{align} \frac{(e-1)(e-2)}{2}=g\leq m(m-1+\varepsilon)= \frac{e-\varepsilon}{2}\cdot \frac{e+\varepsilon-2}{2}. \end{align} $$

Hence, we get a contradiction, as inequality (3.5) fails for any $\varepsilon \in \{0,1\}$ and $e\geq 4$ .

3.2 Proof of Theorem 1.1

In this subsection, we prove Theorem 1.1, and we briefly discuss the covering gonality of $C^{(k)}$ when $0\leq g\leq k$ .

Proof of Theorem 1.1.

Let $k\in \{3,4\}$ and let C be a smooth curve of genus $g\geq k+1$ , with $\big (g,\operatorname {\mathrm {gon}}(C)\big )\neq (k+1,k)$ . In light of (3.3), we need to prove that $\operatorname {\mathrm {cov.gon}}\big (C^{(k)}\big )\geq \operatorname {\mathrm {gon}}(C)$ .

Since $C^{(k)}$ is a variety of general type, it is not covered by rational curves. It follows that $\operatorname {\mathrm {cov.gon}}\big (C^{(k)}\big )\geq 2$ , and the assertion holds when C is a hyperelliptic curve.

We assume hereafter that C is non-hyperelliptic and, in order to simplify notation, we identify the curve with its canonical model $\phi (C)\subset \mathbb {P}^{g-1}$ .

Aiming for a contradiction, we suppose that there exists a covering family $\mathcal {E}\stackrel {\pi }{\longrightarrow } T$ of d-gonal curves, with $d<\operatorname {\mathrm {gon}}(C)$ . Therefore, as $\operatorname {\mathrm {gon}}(C)\leq \left\lfloor \frac {g+3}{2}\right\rfloor $ (see, for example, [1, Theorem V.1.1]), we have

(3.6) $$ \begin{align} d\leq \operatorname{\mathrm{gon}}(C)-1\leq \left\lfloor\frac{g+1}{2}\right\rfloor \quad\text{and}\quad g\geq 2d-1. \end{align} $$

Using notation as in Definition 3.1 and (3.1), we consider a general point $(t,y)\in T\times \mathbb {P}^1$ and its fiber $\varphi _t^{-1}(t,y)=\left\{x_1,\ldots ,x_d\right\}\subset E_t$ via the d-gonal map $\varphi _t\colon E_t \longrightarrow \{t\}\times \mathbb {P}^1$ . Moreover, for any $i=1,\dots ,d$ , we define $P_i:=f_t(x_i)\in C^{(k)}$ and we set

(3.7) $$ \begin{align} P_1=p_1+\dots+p_{k} \quad P_2=p_{k+1}+\dots+p_{2k} \quad \dots\, \quad P_d=p_{(d-1)k+1}+\dots+p_{dk}. \end{align} $$

We point out that, since $(t,y)\in T\times \mathbb {P}^1$ is general, the points $P_i\in C^{(k)}$ are distinct, and they lie outside the diagonal divisor of $C^{(k)}$ (i.e. for any $i=1,\dots ,d$ , the points $p_{(i-1)k+1},\dots ,p_{ik}\in C$ are distinct). Moreover, for $i=1,\dots ,d$ , the linear space

(3.8) $$ \begin{align} \Lambda_i:=\operatorname{\mathrm{Span}}\left(p_{(i-1)k+1},\dots,p_{ik}\right)\subset \mathbb{P}^{g-1} \end{align} $$

spanned by $\operatorname {\mathrm {Supp}} (P_i)$ has dimension $k-1$ , and it is parameterized by $\gamma (P_i)\in \mathbb {G}(k-1,g-1)$ . By Theorem 3.2, the $(k-1)$ -planes $\Lambda _1,\dots ,\Lambda _d$ are in special position with respect to $(g-1-k)$ -planes. In particular, the dimension of the linear span of the points $p_1,\dots ,p_{dk}$ is governed by Corollary 2.6.

Let us consider the effective divisor $D=D_{(t,y)}\in \operatorname {Div}(C)$ given by

(3.9) $$ \begin{align} D:=p_1+\dots +p_{dk}=\sum_{j=1}^h n_jq_j, \end{align} $$

where the points $q_j\in \{p_1,\dots ,p_{dk}\}$ are assumed to be distinct and $n_j=\operatorname {\mathrm {mult}}_{q_j}(D)$ . We now distinguish some cases depending on the values of the integers $n_j$ .

Case A: suppose that $n_j=1$ for any $j=1,\dots ,h$ – that is, the points $p_1,\dots ,p_{dk}$ are all distinct.

Case A.1: assume in addition that $d=2$ (i.e., the map $\varphi _t\colon E_t \longrightarrow \{t\}\times \mathbb {P}^1$ is hyperelliptic). Since the $(k-1)$ -planes $\Lambda _1$ and $\Lambda _2$ are $\operatorname {\mathrm {SP}}(g-1-k)$ , they coincide – that is, $\gamma (P_1)=\gamma (P_2)$ (cf. Remark 2.3). In particular, $\dim \operatorname {\mathrm {Span}}(p_1,\dots ,p_{2k})=k-1$ , and the geometric version of the Riemann–Roch theorem (see [1, p. 12]) yields

(3.10) $$ \begin{align} \dim |D|=\deg D-1-\dim\operatorname{\mathrm{Span}}(p_1,\dots,p_{2k})=2k-1-(k-1)=k=\frac{\deg D}{2}. \end{align} $$

Since $D\neq 0$ and C is assumed to be non-hyperelliptic, Clifford’s theorem (cf. [1, p. 107]) assures that D is a canonical divisor of C, and hence, $g=k+1$ . We note that for $k=3$ , we obtain $g=4$ and $2=d<\operatorname {\mathrm {gon}}(C)\leq 3$ , but the case $\big (g,\operatorname {\mathrm {gon}}(C)\big )= (4,3)$ is excluded by assumption.

Thus, we only need to consider the case where $k=4$ and $g=5$ . Let $\iota \colon C^{(4)}\dashrightarrow C^{(4)}$ be the (rational) involution $y_1+\dots +y_4 \longmapsto \operatorname {\mathrm {Span}}(y_1,\dots ,y_4)|_C-(y_1+\dots +y_4)$ sending a general point to the residual of the canonical divisor $\operatorname {\mathrm {Span}}(y_1,\dots ,y_4)|_C\in \operatorname {Div}(C)$ cut out on C by the hyperplane $\operatorname {\mathrm {Span}}(y_1,\dots ,y_4)\subset \mathbb {P}^4$ . Since $\Lambda _1=\operatorname {\mathrm {Span}}(p_1,\dots ,p_4)$ is a hyperplane and $D=P_1+P_2$ is a canonical divisor, we deduce that $\operatorname {\mathrm {Span}}(p_1,\dots ,p_4)|_C=D$ , and hence, $\iota (P_1)=D-P_1=P_2$ . Therefore, as $\varphi _t^{-1}(t,y)=\left\{x_1,x_2\right\}\subset E_t$ is a general fiber of the hyperelliptic map and $P_i=f(x_i)$ , the restriction of $\iota $ to the curve $f(E_t)\subset C^{(4)}$ is the automorphism induced by the hyperelliptic involution of $E_t$ .

By Riemann’s theorem (see, for example) [1, p. 27]), the composition of the Abel-Jacobi map with a suitable translation gives a morphism $v\colon C^{(4)}\longrightarrow J(C)$ , which maps birationally to the theta divisor $\Theta \subset J(C)$ . Furthermore, the $(-1)$ –involution on the Jacobian variety $J(C)$ restricts to an involution on $\Theta $ , which satisfies $-v(P)=v(K_C-P)$ for any canonical divisor $K_C$ on C such that $K_C-P\geq 0$ (cf. [1, p. 27]). Since for general $P\in C^{(4)}$ the divisor $P+\iota (P)$ is canonical, we obtain $-v(P)=v(\iota (P))$ – that is, we have a commutative diagram

Let us consider the quotient map $\psi \colon \Theta \longrightarrow \Theta /\langle -1\rangle \subset \operatorname {\mathrm {Kum}} (C)$ , where $\operatorname {\mathrm {Kum}}(C)=J(C)/\langle -1\rangle $ is the Kummer variety. Since $\iota $ acts on $f(E_t)\subset C^{(4)}$ as the hyperelliptic involution, the fourfold $\Theta /\langle -1\rangle $ is covered by rational curves $(\psi \circ v)\left(f(E_t)\right)$ .

We recall that the morphisms $C^{(4)}\stackrel {v}{\longrightarrow }J(C)\longrightarrow \operatorname {\mathrm {Kum}}(C)$ induce a chain of isomorphisms on $(4,0)$ -forms

(3.11) $$ \begin{align} H^{4,0}\left(C^{(4)}\right)\cong H^{4,0}\left(J(C)\right)\cong \bigwedge^4 H^0(C,\omega_C) \cong H^{4,0}\left(\operatorname{\mathrm{Kum}}(C)\right), \end{align} $$

where the last isomorphism occurs because the action of the $(-1)$ –involution on $H^{1,0}\left(J(C)\right)\cong H^0(C,\omega _C)$ is anti-invariant, so the action on $\bigwedge ^4 H^0(C,\omega _C)$ is invariant (cf. e.g., [24]). Let Y be a desingularization of $ \Theta /\langle -1\rangle $ , and let X be a suitable blow-up of $C^{(4)}$ such that we have a commutative diagram

(3.12)

By (3.11) and (3.12), we obtain a morphism $X\longrightarrow \operatorname {\mathrm {Kum}}(C)$ such that the map induced on $(4,0)$ -forms is an isomorphism, which factors through $H^{4,0}\left(Y\right)$ . In particular, $H^{4,0}\left(C^{(4)}\right) \cong H^{4,0}\left(X\right)= \eta ^* H^{4,0}\left(Y\right)$ . Thus, the canonical map of X factors through the canonical map of Y. Moreover, these maps are generically finite because the canonical map of $C^{(4)}$ is. Hence, Y is a variety of general type, and it cannot be covered by rational curves. So we get a contradiction, as Y is birational to $\Theta /\langle -1\rangle $ , which is covered by rational curves.

Case A.2: assume that $d\geq 3$ . Consider the sequence $\Lambda _1,\dots ,\Lambda _d\subset \mathbb {P}^{g-1}$ of $(k-1)$ -planes satisfying $\operatorname {\mathrm {SP}}(g-1-k)$ , together with a partition of the set of indices $\left\{1,\dots ,d\right\}$ such that each part $\{i_1,\dots ,i_t\}$ corresponds to an indecomposable sequence $\Lambda _{i_1},\dots ,\Lambda _{i_t}$ of $(k-1)$ -planes that satisfy $\operatorname {\mathrm {SP}}(g-1-k)$ , as in the assumption of Corollary 2.6.

We point out that each part consists of at least $3$ elements. Indeed, if we had two planes $\Lambda _{i_1}$ and $\Lambda _{i_2}$ in special position, we could argue as in Case A.1 and we would deduce that $g=k+1$ . However, this is impossible, because we know that $3\leq d\leq \operatorname {\mathrm {gon}}(C)-1\leq \left\lfloor \frac {g+1}{2}\right\rfloor $ , with $k\in \{3,4\}$ and $\big (k,g,\operatorname {\mathrm {gon}}(C)\big )\neq (4,5,4)$ by assumption.

Let m be the number of parts. Hence, $m\leq \left\lfloor \frac {d}{3}\right\rfloor $ and Corollary 2.6 implies

$$ \begin{align*} \dim\operatorname{\mathrm{Span}}(p_1,\dots,p_{dk}) & =\dim\operatorname{\mathrm{Span}}\left(\Lambda_1,\dots,\Lambda_d\right)\leq d+k-3 +(m-1)(k-2) \notag \\ &\leq d+k-3+\left(\left\lfloor\frac{d}{3}\right\rfloor-1\right)(k-2). \end{align*} $$

By the geometric version of the Riemann–Roch theorem, we obtain

(3.13) $$ \begin{align} \dim |D|&=\deg D-1-\dim\operatorname{\mathrm{Span}}(p_1,\dots,p_{dk})\geq dk-1-\left(d+k-3+\left(\left\lfloor\frac{d}{3}\right\rfloor-1\right)(k-2)\right) \notag\\ & = (k-1)d -(k-2)\left\lfloor\frac{d}{3}\right\rfloor>\frac{kd}{2}=\frac{\deg D}{2}, \end{align} $$

where the last inequality holds because $k\geq 3$ . Hence, Clifford’s theorem assures that $\deg D\geq 2g$ , so that $\dim |D|=\deg D-g=kd-g$ by the Riemann–Roch theorem. Combining this fact and (3.13), we obtain $kd-g\geq (k-1)d -(k-2)\frac {d}{3}$ , which gives $3g\leq d(k+1)$ . Then we conclude by (3.6) that $(k,g,d)=(4,5,3)$ and, in particular, $\operatorname {\mathrm {gon}}(C)=4$ . However, $\big (k,g,\operatorname {\mathrm {gon}}(C)\big )\neq (4,5,4)$ by assumption, so we get a contradiction.

Case B: suppose that the points $p_1,\dots ,p_{kd}$ are not distinct (i.e., the integers $n_j$ are not all equal to  $1$ ). Let us denote by

(3.14) $$ \begin{align} N_{\alpha}:=\left\{\left. q_j\in\operatorname{\mathrm{Supp}}(D)\right|n_j=\alpha\right\} \end{align} $$

the set of the points supporting D such that $\operatorname {\mathrm {mult}}_{q_j}D=\alpha $ . Given some $\alpha \geq 1$ such that $N_{\alpha }\neq \emptyset $ , we may assume $N_{\alpha }=\{q_1,\dots ,q_s\}$ , with $s\leq \left\lfloor \frac {\deg D}{\alpha }\right\rfloor =\left\lfloor \frac {kd}{\alpha }\right\rfloor $ .

Since $T\times \mathbb {P}^1$ is irreducible, there exists a suitable open subset $U\subset T\times \mathbb {P}^1$ such that, as we vary $(t,y)\in U$ , the multiplicities $n_j$ of the points supporting the divisor $D=D_{(t,y)}$ given by (3.9) do not vary (i.e., the cardinality of each $N_{\alpha }$ is constant on U). Thus, we can define a rational map $\xi _{\alpha }\colon T\times \mathbb {P}^1\dashrightarrow C^{(s)}$ sending a general point $(t,y)\in U$ to the effective divisor $Q:=q_1+\dots +q_s\in C^{(s)}$ such that $N_{\alpha }=\{q_1,\dots ,q_s\}$ .

For a general $t\in T$ , let $\xi _{\alpha ,t}\colon \{t\}\times \mathbb {P}^1\dashrightarrow C^{(s)}$ be the restriction of $\xi _{\alpha }$ to $\{t\}\times \mathbb {P}^1$ , and let us consider the composition with the Abel–Jacobi map u,

As the Jacobian variety $J(C)$ does not contain rational curves, the map $u\circ \xi _{\alpha ,t}$ must be constant.

Case B.1: suppose that for any $\alpha $ such that $N_{\alpha }\neq \emptyset $ , the map $\xi _{\alpha ,t}\colon \{t\}\times \mathbb {P}^1\dashrightarrow C^{(s)}$ is non-constant. It follows from Abel’s theorem that $|q_1+\dots +q_s|$ is a $\mathfrak {g}^r_s$ containing $\xi _{\alpha ,t}\left(\{t\}\times {\mathbb {P}}^1\right)$ , so that $r\geq 1$ and $s=|N_{\alpha }|\geq \operatorname {\mathrm {gon}}(C)$ . Hence, $\alpha \leq k-1$ because, otherwise, we would have $s\leq \left\lfloor \frac {kd}{\alpha }\right\rfloor \leq d<\operatorname {\mathrm {gon}}(C)$ .

We also point out that $\alpha |N_{\alpha }|=\alpha s$ must be a multiple of d. To see this fact, let $i=1,\dots ,d$ and consider the points $P_i=p_{(i-1)k+1}+\dots +p_{ik}$ in (3.7). Then the number of points $p_j\in N_{\alpha }\cap \operatorname {\mathrm {Supp}}(P_i)$ does not depend on i; otherwise, by varying $(t,y)\in \{t\}\times \mathbb {P}^1$ , the points $P_i$ would describe different irreducible components of $E_t$ , but such a curve is irreducible. Hence, $\alpha |N_{\alpha }|$ – which is the number of points among $p_1,\dots ,p_{dk}$ contained in $N_{\alpha }$ – must be a multiple of d.

We claim that $N_1=\emptyset $ and there exists a unique $2\leq \alpha \leq k-1$ such that $N_{\alpha }\neq \emptyset $ , with in particular $|N_{\alpha }|=\frac {kd}{\alpha }$ . Indeed, if $k=3$ , we have that $3d=\deg (D)=|N_1|+2|N_2|$ . Then there exists $j\in \{1,2\}$ such that $|N_j|\leq d$ , but we observed that if $N_j\neq \emptyset $ , then $|N_j|\geq \operatorname {\mathrm {gon}}(C)>d$ . By assumption of Case B, we have $N_2\neq \emptyset $ , so we conclude that $N_1$ is empty and $|N_2|=\frac {3d}{2}$ . Similarly, if $k=4$ and $N_3\neq \emptyset $ , then $|N_3|>d$ and $4d=\deg D=|N_1|+2|N_2|+3|N_3|$ , so that $N_1=N_2=\emptyset $ and $|N_3|=\frac {4d}{3}$ . Finally, if $k=4$ and $N_3= \emptyset $ , then $N_2\neq \emptyset $ and $4d=\deg D=|N_1|+2|N_2|$ . Since $|N_2|>d$ and $2|N_2|$ is a multiple of d, we conclude that $N_1=\emptyset $ and $|N_2|=2d$ .

Accordingly, we have to discuss the following three cases:

1. (i) $k=3$ and $N_2\neq \emptyset $ ;

2. (ii) $k=4$ and $N_3\neq \emptyset $ ;

3. (iii) $k=4$ and $N_2\neq \emptyset $ .

To this aim, we use the following fact.

Claim 3.6. Consider the points $P_1,\dots ,P_d$ in (3.7). Let $A\subset \{q_1,\ldots ,q_s\}$ be a set of points such that for some $j\in \{1,\dots ,d\}$ , we have $A\cap \operatorname {\mathrm {Supp}}(P_j)=\emptyset $ and $A\cap \operatorname {\mathrm {Supp}}(P_i)\neq \emptyset $ for any $i\neq j$ . Then $|A|\geq \operatorname {\mathrm {gon}}(C)-k$ .

Proof of Claim 3.6.

Consider the $(k-1)$ -planes $\Lambda _1,\dots ,\Lambda _d$ defined in (3.8) and the linear space $\operatorname {\mathrm {Span}}(A)\subset \mathbb {P}^{g-1}$ , where $\dim \operatorname {\mathrm {Span}}(A)\leq |A|-1$ . Of course, if $\dim \operatorname {\mathrm {Span}}(A)> g-1-k$ , then $\operatorname {\mathrm {Span}}(A)$ intersects the $(k-1)$ -plane $\Lambda _j$ . If instead $\dim \operatorname {\mathrm {Span}}(A)\leq g-1-k$ , then $\operatorname {\mathrm {Span}}(A)$ meets $\Lambda _j$ because $\Lambda _1,\dots ,\Lambda _d$ are $\operatorname {\mathrm {SP}}(g-1-k)$ and $A\cap \Lambda _i\neq \emptyset $ for any $i=1,\dots ,\widehat {j},\dots ,d$ . In any case, $\operatorname {\mathrm {Span}}(A)\cap \Lambda _j\neq \emptyset $ , so that the space $\operatorname {\mathrm {Span}}(A,\Lambda _j)$ has dimension at most $|A|+k-2$ and contains (at least) $|A|+k$ distinct points of C (the elements of A and the k points supporting $P_j$ ). By the geometric version of the Riemann–Roch theorem, those points define a $\mathfrak {g}^r_{|A|+k}$ on C, with $r\geq 1$ . Thus, $|A|+k\geq \operatorname {\mathrm {gon}}(C)$ .

(i) If $k=3$ and $N_2\neq \emptyset $ , the divisor D has the form $D=2(q_1+\dots +q_s)$ , with $s=\frac {3d}{2}$ . We note that $d\neq 2$ ; otherwise, we would have $P_1=P_2=q_1+q_2+q_3$ . As s is an integer, we may set $d=2c$ and $s=3c$ , for some integer $c\geq 2$ .

Suppose that $c\neq 2$ . Given a point $x\in \{q_1,\dots ,q_s\}\subset C$ , it belongs to the support of two points of $C^{(3)}$ , say $P_1$ and $P_2$ . Then $\operatorname {\mathrm {Supp}}(P_1)\cup \operatorname {\mathrm {Supp}}(P_2)$ consists of at most 5 distinct points. As $s=3c\geq 9$ , there exists a point $y\in \{q_1,\dots ,q_s\}\setminus \{x\}$ belonging to the support of two other points of $C^{(3)}$ , say $P_3$ and $P_4$ . Now, we can choose $n\leq d-5$ distinct points $a_1,\dots ,a_n\in \{q_1,\dots ,q_s\}\setminus \{x,y\}$ , such that the set $A:=\left\{x,y,a_1,\dots ,a_n\right\}$ contains a point in the support of $P_i$ for any $i=1,\dots ,d-1$ and $A\cap \operatorname {\mathrm {Supp}}(P_d)=\emptyset $ (given x and y, it suffices to choose – at most – one point in the support of any $P_5,\ldots , P_{d-1}$ , avoiding the points of $\operatorname {\mathrm {Supp}}(P_d)$ ). Thus, the set A satisfies the assumption of Claim 3.6, but $|A|\leq d-3<\operatorname {\mathrm {gon}}(C)-3$ , a contradiction.

However, suppose that $c=2$ . Since the curve $E_t$ is irreducible, the points $P_i$ in (3.7) must be indistinguishable. Then it is easy to check that, up to reordering indices, there are only two admissible configurations:

1. 1. $P_1=q_1+q_2+q_3$ , $P_2=q_1+q_4+q_5$ , $P_3=q_2+q_4+q_6$ , $P_4=q_3+q_5+q_6$ ;

2. 2. $P_1=q_1+q_2+q_3$ , $P_2=q_1+q_2+q_4$ , $P_3=q_5+q_6+q_3$ , $P_4=q_5+q_6+q_4$ .

In both cases, we have that the line $\operatorname {\mathrm {Span}}(q_4,q_5)$ meets the planes $\Lambda _2$ , $\Lambda _3$ , and $\Lambda _4$ . Since $\Lambda _1,\dots ,\Lambda _4$ are $\operatorname {\mathrm {SP}}(g-4)$ , we deduce that $\Lambda _1\cap \operatorname {\mathrm {Span}}(q_4,q_5)\neq \emptyset $ , so that $\operatorname {\mathrm {Span}}(\Lambda _1,q_4,q_5)= \operatorname {\mathrm {Span}}(q_1,\dots ,q_5)\cong \mathbb {P}^3$ . Analogously, by considering the line $\operatorname {\mathrm {Span}}(q_4,q_6)$ , we deduce that $\operatorname {\mathrm {Span}}(q_1,\dots ,q_4,q_6)\cong \mathbb {P}^3$ . Therefore, either $\operatorname {\mathrm {Span}}(q_1,\dots ,q_4)\cong \mathbb {P}^2$ or $\operatorname {\mathrm {Span}}(q_1,\dots ,q_6)\cong \mathbb {P}^3$ . In the former case, $|q_1+\dots +q_4|$ is a $\mathfrak {g}^1_4$ on C, which is impossible as $\operatorname {\mathrm {gon}}(C)>d=2c=4$ .

In the latter case, we set $L:=q_1+\dots +q_6$ , and $|L|$ turns out to be a $\mathfrak {g}^2_6$ on C. Since $\operatorname {\mathrm {gon}}(C)>4$ , $\dim |L-p-q|=0$ for any $p,q\in C$ , so that $|L|$ is very ample. Hence, $|L|$ is the unique $\mathfrak {g}^2_6$ on C (see, for example, [10, Teorema 3.14]). Therefore, by retracing our construction, we have that if $P'\in C^{(3)}$ is a general point, then there exists an effective divisor $L'=q^{\prime }_1+\dots +q^{\prime }_s$ such that $|L'|=|L|$ and $P'\leq L'$ . Thus, we obtain a contradiction as $C^{(3)}$ is a threefold, but $\dim |L|=2$ .

In conclusion, case (i) does not occur.

(ii) If $k=4$ and $N_3\neq \emptyset $ , we have $D=3(q_1+\dots +q_s)$ , with $s=\frac {4d}{3}$ . In particular, each $q_j$ belongs to the support of three distinct points among $P_1,\dots ,P_d\in C^{(4)}$ . We argue as above, and we set $d=3c$ and $s=4c$ , for some integer $c\geq 2$ (if c equaled 1, then we would have $P_1=P_2=P_3=q_1+\dots +q_4$ ).

Suppose that $c\neq 2$ . Given a point $x\in \{q_1,\dots ,q_s\}$ , it belongs to the support of three points of $C^{(4)}$ , say $P_1,P_2,P_3$ . Then $\operatorname {\mathrm {Supp}}(P_1)\cup \operatorname {\mathrm {Supp}}(P_2)\cup \operatorname {\mathrm {Supp}}(P_3)$ consists of at most 10 distinct points. As $s=4c\geq 12$ , there exists a point $y\in \{q_1,\dots ,q_s\}\setminus \{x\}$ belonging to the support of three other points of $C^{(4)}$ , say $P_4,P_5,P_6$ . Hence, we can choose $n\leq d-7$ distinct points $a_1,\dots ,a_n\in \{q_1,\dots ,q_s\}\setminus \{x,y\}$ , such that the set $A:=\left\{x,y,a_1,\dots ,a_n\right\}$ contains a point in the support of $P_i$ for any $i=1,\dots ,d-1$ and $A\cap \operatorname {\mathrm {Supp}}(P_d)=\emptyset $ . Therefore, A satisfies the assumption of Claim 3.6, with $|A|\leq d-5<\operatorname {\mathrm {gon}}(C)-4$ , a contradiction.

If instead $c=2$ , we have $d=6$ and $s=8$ . In this setting, we can choose two points $x,y\in \{q_1,\dots ,q_8\}$ such that $x+y\leq P_i$ for only one $i\in \{1,\dots ,d\}$ . To see this fact, consider the point $q_1$ and, without loss of generality, suppose that $q_1$ belongs to the support of $P_1,P_2,P_3$ and $\operatorname {\mathrm {Supp}}(P_1+P_2+P_3)=\left\{q_1,\dots ,q_r\right\}$ for some $5\leq r\leq 8$ . If there exists $j\in \{2,\dots ,r\}$ such that $q_j$ lies on the support of only one point among $P_1,P_2,P_3$ , then $q_1+q_j\leq P_i$ for only one $i\in \{1,2,3\}$ , and we may set $(x,y)=(q_1,q_j)$ . If instead any $q_j$ belongs to the support of two points among $P_1,P_2,P_3$ , then it is easy to check that $\operatorname {\mathrm {Supp}}(P_1+P_2+P_3)=\left\{q_1,\dots ,q_5\right\}$ because the divisor $P:=P_1+P_2+P_3$ has degree 12, where $\operatorname {\mathrm {mult}}_{p_1}P=3$ and $\operatorname {\mathrm {mult}}_{p_j}P\geq 2$ for any $j=2,\dots , r$ . Then we consider the point $q_6$ , and, without loss of generality, we suppose that $q_6\in \operatorname {\mathrm {Supp}}(P_4)$ . As $s=8$ , $\operatorname {\mathrm {Supp}}(P_4)$ contains at least an element of $\left\{q_2,\dots ,q_5\right\}$ , say $q_2$ . Since $q_2$ belongs also to the support of two points among $P_1,P_2,P_3$ , we conclude that $q_2+q_6\leq P_i$ only for $i=4$ , and we may set $(x,y)=(q_2,q_6)$ .

We point out that $A:=\{x,y\}$ intersects all the sets $\operatorname {\mathrm {Supp}}(P_1),\dots ,\operatorname {\mathrm {Supp}}(P_d)$ but one because x belongs to the support $P_i$ and two other points of $C^{(4)}$ – say $P_2$ and $P_3$ – and $y\in \operatorname {\mathrm {Supp}}(P_i)$ belongs to the support of two points other than $P_2$ and $P_3$ . Hence, A satisfies the assumption of Claim 3.6, with $|A|=2= d-4<\operatorname {\mathrm {gon}}(C)-4$ , a contradiction.

It follows that case (ii) does not occur.

(iii) Finally, let $k=4$ and $N_2\neq \emptyset $ . Then the divisor D has the form $D=2(q_1+\dots +q_s)$ , with $s=2d$ . In particular, each $q_j$ belongs to the support of two distinct points among $P_1,\dots ,P_d\in C^{(4)}$ . We note that $d\neq 2$ ; otherwise, we would have $P_1=P_2=q_1+\dots +q_4$ .

If $d\geq 8$ , we argue as in cases (i) and (ii). Namely, we consider a point $x\in \{q_1,\dots ,q_s\}$ belonging to the support of two points of $C^{(4)}$ , say $P_1$ and $P_2$ , where $\operatorname {\mathrm {Supp}}(P_1)\cup \operatorname {\mathrm {Supp}}(P_2)$ consists of at most 7 elements of $\{q_1,\dots ,q_s\}$ . Moreover, as $s=2d\geq 16$ , up to reorder indices, we may consider two distinct points $y,z\in \{q_1,\dots ,q_s\}\setminus \{x\}$ such that $y\in \operatorname {\mathrm {Supp}}(P_3)\cup \operatorname {\mathrm {Supp}}(P_4)$ and $z\in \operatorname {\mathrm {Supp}}(P_5)\cup \operatorname {\mathrm {Supp}}(P_6)$ . Then we can choose $n\leq d-7$ distinct points $a_1,\dots ,a_n\in \{q_1,\dots ,q_s\}\setminus \{x,y,z\}$ , such that the set $A:=\left\{x,y,z,a_1,\dots ,a_n\right\}$ contains a point in the support of $P_i$ for any $i=1,\dots ,d-1$ and $A\cap \operatorname {\mathrm {Supp}}(P_d)=\emptyset $ . Thus, A satisfies the assumption of Claim 3.6, but $|A|\leq d-4<\operatorname {\mathrm {gon}}(C)-4$ , a contradiction.

If $3\leq d\leq 7$ , we consider the divisor $L:=D_{\text {red}}=q_1+\dots +q_{2d}\in \operatorname {Div}(C)$ . When $d= 3$ , we have that $\operatorname {\mathrm {Span}}(q_1,\dots ,q_{6})$ has dimension 3. Indeed, any point $q_j$ lies on two $3$ -planes among $\Lambda _1,\Lambda _2,\Lambda _3$ , and by special position property, it must belong to the third one too. Thus, $q_1,\dots ,q_{6}$ span a $3$ -plane and L is a complete $\mathfrak {g}^2_6$ . Hence, $|L-q_6|$ is a complete $\mathfrak {g}^r_5$ , where $1\leq r\leq 2$ and $L-q_6=P_1+q_5$ . We note further that $\dim W^1_5\leq 2$ and $\dim W^2_5\leq 0$ by Martens’ theorem (see [1, Theorem IV.5.1]). Then we argue similarly to case (i). Namely, by retracing our construction, we have that if $P'\in C^{(4)}$ is a general point, then there exists and effective divisor $L'=q^{\prime }_1+\dots +q^{\prime }_{5}$ such that $P'\leq L'$ and $|L'|$ is a complete $\mathfrak {g}^{r}_{5}$ , with $1\leq r\leq 2$ . Therefore, we get a contradiction since $\dim C^{(4)}=4$ , whereas the locus of divisors $L'\in C^{(5)}$ as above has dimension $\dim |L'|\leq 3$ .

As far as the remaining values of d are concerned, we need the following.

Claim 3.7. If $4\leq d\leq 7$ , then $\dim \operatorname {\mathrm {Span}}(q_1,\dots ,q_s)\leq d$ .

Proof of Claim 3.7.

It follows from (3.6) that $g-1\geq 2d-2> 4$ . Hence, the General Position Theorem ensures that any 3-plane $\Lambda _i$ is exactly 4-secant to C, and meets C (transversally) only at $\operatorname {\mathrm {Supp}}(P_i)$ . Therefore, without loss of generality, we may assume that $q_1\in \Lambda _1\cap \Lambda _2$ and $q_1\not \in \Lambda _{3},\dots ,\Lambda _{d}$ . Let $\pi _1\colon \mathbb {P}^{g-1}\dashrightarrow {\mathbb {P}}^{g-2}$ be the projection from $q_1$ . By Lemma 2.13, for $i=3,\dots ,d$ , the $3$ -planes $\Gamma _i:=\pi _1(\Lambda _i)$ are in special position with respect to the $(g-6)$ -planes.

Suppose that the sequence $\Gamma _3,\dots ,\Gamma _d$ is decomposable in the sense of Definition 2.4. This is possible only if $6\leq d\leq 7$ , and there exists a part with $2$ elements, say $\{3,4\}$ . Hence, $\Gamma _3=\Gamma _4\cong \mathbb {P}^3$ , and $\overline {\pi _1^{-1}(\Gamma _3)}=\operatorname {\mathrm {Span}}(q_1,\Lambda _3,\Lambda _4)\cong \mathbb {P}^4$ contains at least 6 points of C. Thus, C admits a $\mathfrak {g}^1_6$ , which is impossible as $\operatorname {\mathrm {gon}}(C)>d\geq 6$ .

So the sequence $\Gamma _3,\dots ,\Gamma _d$ is indecomposable, and Theorem 2.5 ensures that $\dim \operatorname {\mathrm {Span}}(\Gamma _3,\dots ,\Gamma _d)\leq d-1$ . It follows that $H_1:=\operatorname {\mathrm {Span}}(q_1,\Lambda _3,\dots ,\Lambda _d)$ satisfies $\dim H_1\leq d$ . If $q_j\in H_1$ for any $q_j\in \Lambda _1\cap \Lambda _2$ , the claim is proved.

By contradiction, suppose that there exists $q_j\in \Lambda _1\cap \Lambda _2$ such that $q_j\not \in H_1$ . Then we can consider the projection $\pi _j\colon \mathbb {P}^{g-1}\dashrightarrow {\mathbb {P}}^{g-2}$ , and, by arguing as above, we obtain that $H_j:=\operatorname {\mathrm {Span}}(q_j,\Lambda _3,\dots ,\Lambda _d)$ satisfies $\dim H_j\leq d$ . Since $q_j\not \in H_1$ , we deduce that $H:=\operatorname {\mathrm {Span}}(\Lambda _3,\dots ,\Lambda _d)=H_1\cap H_j$ has dimension $\dim H\leq d-1$ .

If $d=4$ , this implies $\Lambda _3=\Lambda _4$ , which is impossible because $\Lambda _3$ does not contain all the points of $\operatorname {\mathrm {Supp}}(P_4)$ .

If $d=5$ , Lemma 2.12 yields that $\Lambda _3,\Lambda _4,\Lambda _5$ are $\operatorname {\mathrm {SP}}(g-5)$ because $p_1\not \in H$ and $\Gamma _3,\Gamma _4,\Gamma _5$ are $\operatorname {\mathrm {SP}}(g-6)$ . Since $s=10$ , there exists a point $q_j$ in the support of two points among $P_3,P_4,P_5$ . Hence, $q_j$ lies on two of the spaces $\Lambda _3,\Lambda _4,\Lambda _5$ , and by special position property, it has to lie also on the third one, but this is impossible.

If $d=6$ , then H contains at least $2(d-2)=8$ points in $\{q_1,\dots ,q_{12}\}$ because for $i=3,\dots ,6$ , each space $\Lambda _i$ contains exactly 4 points and each point may lie only on two $3$ -planes among $\Lambda _1,\dots ,\Lambda _6$ . Moreover, if H contained exactly 8 points, then both $\Lambda _1$ and $\Lambda _2$ should contain the remaining 4, a contradiction. Thus, H contains at least 9 points of C, and as $\dim H\leq 5$ , we have that C admits a $\mathfrak {g}^r_9$ with $r\geq 3$ . Since $\operatorname {\mathrm {gon}}(C)>d=6$ , the $\mathfrak {g}^r_9$ satisfies the assumption of Lemma 3.5. Hence, we obtain a contradiction as $r\neq 2$ .

If $d=7$ , we argue as in the previous case, and we note that H contains at least $2(d-2)=10$ points in $\{q_1,\dots ,q_{14}\}$ , with $\dim H\leq 6$ . Then C admits a $\mathfrak {g}^r_{10}$ with $r\geq 3$ , and since $\operatorname {\mathrm {gon}}(C)>d=7$ , Lemma 3.5 leads to a contradiction.

For any $4\leq d\leq 7$ , Claim 3.7 ensures that the divisor $L=q_1+\dots +q_{2d}$ defines a linear series $|L|$ of dimension $r= \deg L-1-\dim \operatorname {\mathrm {Span}}(q_1,\dots ,q_{2d})\geq 2d-1-d=d-1\geq 3$ . By combining (3.4) and Claim 3.7, we then obtain

$$ \begin{align*}d-1\leq\operatorname{\mathrm{gon}}(C)-2\leq \dim\operatorname{\mathrm{Span}}(q_1,\dots,q_{2d})\leq d. \end{align*} $$

If $\dim \operatorname {\mathrm {Span}}(q_1,\dots ,q_{2d})=d-1$ , then $|L|$ turns out to be a $\mathfrak {g}^{d}_{2d}$ . However, this is impossible by Clifford’s theorem and (3.6), as $d\geq 4$ . Hence, it follows that $\dim \operatorname {\mathrm {Span}}(q_1,\dots ,q_{2d})=d$ and $|L|$ is a complete $\mathfrak {g}^{d-1}_{2d}$ . Moreover, either L is very ample, or there exist $p,q\in C$ such that $\dim |L-p-q|\geq d-2$ .

Therefore, in order to conclude case (iii), we analyze separately the following three cases: (iii.a) L is very ample, (iii.b) there exists $p\in C$ such that $\dim |L-p|=d-1$ , (iii.c) there exist $p,q\in C$ such that $\dim |L-p|=\dim |L-p-q|=d-2$ .

(iii.a) If L is very ample, we claim that $4\leq d\leq 5$ . If indeed $d\geq 6$ , then $2d-1=m(d-2)+\varepsilon $ with $m=2$ and $\varepsilon =3$ , so that Castelnuovo’s bound would imply $g\leq \frac {m(m-1)}{2}(d-2)+m\varepsilon =d+4$ . Hence, (3.6) implies $2d-1\leq g\leq d+4$ , which is impossible for $d\geq 6$ .

When $d=5$ , $|L|$ is a very ample complete $\mathfrak {g}^{4}_{10}$ . By Castelnuovo’s bound, we obtain $g\leq 9$ . Moreover, as $g\geq 2d-1=9$ , we deduce that C maps isomorphically to an extremal curve of degree $10$ in $\mathbb {P}^4$ (i.e., to a curve whose genus attains the maximum in Castelnuovo’s bound). Thus, [1, Theorem III.2.5 and Corollary III.2.6] ensure that C possesses a $\mathfrak {g}^{1}_{3}$ , but this contradicts the assumption $\operatorname {\mathrm {gon}}(C)>d=5$ .

When $d=4$ , $|L|$ is a very ample complete $\mathfrak {g}^{3}_{8}$ . Let $\phi _{|L|}\colon C\longrightarrow \mathbb {P}^3$ be the corresponding embedding, and consider the Cayley’s number of $4$ -secant lines to a space curve of degree $8$ and genus g, defined as

$$ \begin{align*}C(8,g,3):=\frac{g^2-21g+100}{2}. \end{align*} $$

Thanks to Castelnuovo’s enumerative formula counting $(2r-2)$ -secant $(r-2)$ -planes to irreducible curves in $\mathbb {P}^r$ (see, for example, [13, Theorem 1.2]), we have that if $C(8,g,3)\neq 0$ , then $\phi _{|L|}(C)$ admits at least one $4$ -secant line. As $C(8,g,3)\neq 0$ for any $g\in \mathbb {N}$ , there exists a line $\ell \subset \mathbb {P}^3$ intersecting $\phi _{|L|}(C)$ at $e\geq 4$ points, counted with multiplicities. Thus, the projection from $\ell $ induces a $\mathfrak {g}^{1}_{8-e}$ on $\phi _{|L|}(C)\cong C$ , where $8-e\leq 4= d<\operatorname {\mathrm {gon}}(C)$ , a contradiction.

(iii.b) Suppose that there exists $p\in C$ such that $\dim |L-p|=d-1$ . Then $|L-p|$ is a $\mathfrak {g}^{d-1}_{2d-1}$ , and $\dim |L-p-R|=0$ for any $R\in C^{(d-1)}$ because $\deg (L-p-R)=d<\operatorname {\mathrm {gon}}(C)$ . Therefore, Lemma 3.5 ensures that $d-1=2$ , which contradicts $d\geq 4$ .

(iii.c) Suppose that there exist $p,q\in C$ such that $\dim |L-p|=\dim |L-p-q|=d-2$ . Hence, $|L-p-q|$ is a $\mathfrak {g}^{d-2}_{2d-2}$ , and $\dim |L-p-q-R|=0$ for any $R\in C^{(d-2)}$ because $\deg (L-p-q-R)=d<\operatorname {\mathrm {gon}}(C)$ . Thus, Lemma 3.5 implies that $d=4$ and $|L-p-q|$ is a very ample $\mathfrak {g}^{2}_{6}$ . Therefore, $g=10$ , and $|L|$ is a complete $\mathfrak {g}^{3}_{8}$ .

Of course, C is not a plane quintic, and since $\operatorname {\mathrm {gon}}(C)>d\geq 4$ , C is neither trigonal nor bielliptic. Hence, Mumford’s theorem (see [1, Theorem IV.5.2]) assures that C admits only finitely many complete $\mathfrak {g}^{3}_{8}$ . So we argue similarly to case (i). Namely, by retracing our construction, we have that if $P'\in C^{(4)}$ is a general point, then there exists and effective divisor $L'=q^{\prime }_1+\dots +q^{\prime }_{8}$ such that $P'\leq L'$ and $|L'|$ is a complete $\mathfrak {g}^{3}_{8}$ . Therefore, we get a contradiction since $\dim C^{(4)}=4$ , whereas the locus of divisors $L'\in C^{(8)}$ as above has dimension $\dim |L'|=3$ .

By summing up, we proved that Case B.1 does not occur.

Case B.2: let $\alpha $ such that $N_{\alpha }=\left\{q_1,\dots ,q_s\right\}\neq \emptyset $ and the map $\xi _{\alpha ,t}\colon \{t\}\times \mathbb {P}^1\dashrightarrow C^{(s)}$ is constant. In particular, $Q=q_1+\dots +q_s$ is the only point in the image of $\xi _{\alpha ,t}$ . It follows that for general $y\in \mathbb {P}^1$ , the points $q_1,\dots ,q_s\in C$ are contained in the support of the divisor $D=D_{(t,y)}$ in (3.9). Since the number of points in $N_{\alpha }\cap \operatorname {\mathrm {Supp}}(P_i)$ does not depend on $i=1,\dots ,d$ , the fiber $\varphi _t^{-1}(t,y)=\{x_1,\dots ,x_d\}\subset E_t$ is such that at least one of the points $P_i=f(x_i)$ lies on the $(k-1)$ -dimensional subvariety $q_1+C^{(k-1)}\subset C^{(k)}$ , at least one lies on $q_2+C^{(k-1)}\subset C^{(k)}$ , and so on. Since $(t,y)$ varies on an open subset of $\{t\}\times \mathbb {P}^1$ and $E_t$ is irreducible, we deduce that $f(E_t)$ is contained in the intersection $\bigcap _{j=1}^s \left(q_j+C^{(k-1)}\right)$ , where the points $q_j$ are distinct.

We point out that $s\leq k-2$ . Indeed, if $s\geq k$ , then the intersection above has dimension smaller than 1, so it cannot contain the curve $f(E_t)$ . If instead $s= k-1$ , then $\bigcap _{j=1}^s \left(q_j+C^{(k-1)}\right)$ is the irreducible curve $q_1+\dots +q_{k-1}+C\cong C$ . Hence, we would have $f(E_t)\cong C$ , which is impossible as $\operatorname {\mathrm {gon}}\left(f(E_t)\right)=\operatorname {\mathrm {gon}}(E_t)=d<\operatorname {\mathrm {gon}}(C)$ . Therefore, $s\leq k-2$ and $f(E_t)\subset Q+C^{(k-s)}$ .

Since $k\in \{3,4\}$ , then $1\leq s\leq 2$ , and either $Q=q_1$ or $Q=q_1+q_2$ . In view of the constant map $\xi _{\alpha ,t}\colon \{t\}\times \mathbb {P}^1\dashrightarrow C^{(s)}$ such that $(t,y)\longmapsto Q$ , we may define the map $\phi \colon T\dashrightarrow C^{(s)}$ , sending a general $t\in T$ to the unique point $Q\in C^{(s)}$ in the image of $\xi _{\alpha ,t}$ . As $f(E_t)\subset Q+C^{(k-s)}$ and ${\mathcal {E}}\stackrel {\pi }{\longrightarrow }T$ is a covering family, the map $\phi $ must be dominant, with general fiber $\phi ^{-1}(Q)$ of dimension $\dim T-\dim C^{(s)}=k-s-1$ . Thus, the pullback of ${\mathcal {E}}\stackrel {\pi }{\longrightarrow } T$ to $\phi ^{-1}(Q)\subset T$ (i.e.

$$ \begin{align*}{\mathcal{E}}\times_T \phi^{-1}(Q)\longrightarrow \phi^{-1}(Q))\end{align*} $$

is a $(k-s-1)$ -dimensional family of d-gonal curves $E_t$ , with $f(E_t)\subset Q+C^{(k-s)}$ . Furthermore, the latter family does cover $Q+C^{({k-s})}\cong C^{(k-s)}$ ; otherwise, the whole family ${\mathcal {E}}\stackrel {\pi }{\longrightarrow } T$ would not cover $C^{(k)}$ . Hence, $\operatorname {\mathrm {cov.gon}}\big (C^{(k-s)}\big )\leq \operatorname {\mathrm {gon}}\left(f(E_t)\right)=d<\operatorname {\mathrm {gon}}(C)$ .

If $(k,s)=(3,1)$ or $(k,s)=(4,2)$ , then we obtain $\operatorname {\mathrm {cov.gon}}\big (C^{(2)}\big )<\operatorname {\mathrm {gon}}(C)$ , which contradicts [3, Theorem 1.6]. In particular, this completes the proof of the Theorem 1.1 for $k=3$ , showing that if $g\geq 4$ and $\big (g,\operatorname {\mathrm {gon}}(C)\big )\neq (4,3)$ , then $\operatorname {\mathrm {cov.gon}}(C^{(3)})= \operatorname {\mathrm {gon}}(C)$ .

Finally, if $(k,s)=(4,1)$ , then we have $\operatorname {\mathrm {cov.gon}}(C^{(3)})<\operatorname {\mathrm {gon}}(C)$ , which contradicts the case $k=3$ , and this concludes the proof in the remaining case $k=4$ .

Remark 3.8 (Low genera).

Let C be a smooth curve of genus $0\leq g\leq k$ (i.e. g is outside the range covered by Theorem 1.1), where $g\geq k+1$ . Under this assumption, the k-fold symmetric product $C^{(k)}$ is birational to $J(C)\times \mathbb {P}^{k-g}$ . In particular,

$$ \begin{align*}\operatorname{\mathrm{cov.gon}} \big(C^{(k)}\big)=1\quad\text{for any }0\leq g\leq k-1. \end{align*} $$

When instead $k=g$ , the situation is much more subtle as $C^{(g)}$ is birational to $J(C)$ , and its covering gonality is not completely understood. Since $J(C)$ does not contain rational curves, we deduce that

$$ \begin{align*}2\leq \operatorname{\mathrm{cov.gon}}(J(C))=\operatorname{\mathrm{cov.gon}}\big(C^{(g)}\big) \leq \operatorname{\mathrm{gon}}(C).\end{align*} $$

In particular, we obtain that $\operatorname {\mathrm {cov.gon}}(C^{(2)})=2$ for any curve C of genus $2$ .

When C is a very general curve of genus g, it follows from [23, Theorem 2] that $J(C)$ does not contain hyperelliptic curves if $g=3$ , and [2, Proposition 4] extends this fact to any $g\geq 4$ . In particular, if C is a very general curve of genus $g=3,4$ , then $\operatorname {\mathrm {cov.gon}}(C^{(g)})=\operatorname {\mathrm {gon}}(C)$ , as $\operatorname {\mathrm {gon}}(C)=\left\lfloor \frac {g+3}{2}\right\rfloor =3$ . In this direction, it would be interesting to understand whether the equality $\operatorname {\mathrm {cov.gon}}(C^{(g)})=\operatorname {\mathrm {gon}}(C)$ holds when C is a very general curve of genus $g\geq 5$ .

However, it is worth noticing that equality $\operatorname {\mathrm {cov.gon}}(C^{(g)})=\operatorname {\mathrm {gon}}(C)$ may fail when C is a special curve. For instance, if the curve C is bielliptic and non-hyperelliptic, then $J(C)$ is covered by elliptic curves and $\operatorname {\mathrm {cov.gon}}(C^{(g)})=2<\operatorname {\mathrm {gon}}(C)$ .

Remark 3.9. Concerning other measures of irrationality of an irreducible projective variety X, one could investigate the degree of irrationality $\operatorname {\mathrm {irr}}(X)$ (i.e., the least degree of a dominant rational map $X\dashrightarrow \mathbb {P}^{\dim X}$ ) and the connecting gonality $\operatorname {\mathrm {conn.gon}}(X)$ , that is the least gonality of a curve E passing through two general points $x,y\in X$ , which satisfy $\operatorname {\mathrm {irr}}(X)\geq \operatorname {\mathrm {conn.gon}}(X)\geq \operatorname {\mathrm {cov.gon}}(X)$ .

In [3], various results on $\operatorname {\mathrm {irr}}(C^{(k)})$ are proved. In particular, [3, Theorem 1.3] asserts that if C has very general moduli, then $\operatorname {\mathrm {irr}}(C^{(2)})\geq g-1$ . Unfortunately, we cannot use our techniques in order to obtain a similar result when $k\geq 3$ . Roughly speaking, the problem is that we can no longer ensure by Corollary 2.6 that the linear span $\operatorname {\mathrm {Span}}(p_1,\dots ,p_{dk})$ of the points supporting the divisor D in (3.9) is a proper subspace of $\mathbb {P}^{g-1}$ .

However, if C is a hyperelliptic curve of genus $g\geq 4$ , then $\operatorname {\mathrm {irr}}(C^{(2)})=\big (\operatorname {\mathrm {gon}}(C)\big )^2=4$ by [3, Theorem 1.2]. Furthermore, in the recent paper [8], the authors computed the degree of irrationality of the product $C_1\times C_2$ of two curves $C_i$ of genus $g_i\gg \operatorname {\mathrm {gon}}(C_i)$ , with $i=1,2$ . In particular, they proved that if each $C_i$ is sufficiently general among curves with gonality $\operatorname {\mathrm {gon}}(C_i)$ , then $\operatorname {\mathrm {irr}}(C_1\times C_2)=\operatorname {\mathrm {gon}}(C_1)\cdot \operatorname {\mathrm {gon}}(C_2)$ (cf. [8, Theorem B]), and the same holds when $C_1=C_2$ .

In light of these facts, it is plausible that, by imposing restrictions on the degree of complete linear series on the curve C, one can use the same techniques of this paper to compute the degree of irrationality of the k-fold symmetric product of C, which might agree with bounds analogous to [3, Proposition 1.1 and Remark 6.7].

As far as the connecting gonality of $C^{(k)}$ is concerned, it follows from the results of [7] that if $2\leq k\leq 4$ and C is a general curve of genus $g\geq k+4$ , then the connecting gonality of $C^{(k)}$ is strictly bigger than the covering gonality – that is, $\operatorname {\mathrm {conn.gon}}(C^{(k)})> \operatorname {\mathrm {cov.gon}}(C^{(k)})=\operatorname {\mathrm {gon}}(C)$ (cf. [7, Corollary 1.2]).