Quartic and Quintic Hypersurfaces with Dense Rational Points

Abstract

Let $X_4\subset \mathbb {P}^{n+1}$ be a quartic hypersurface of dimension $n\geq 4$ over an infinite field k. We show that if either $X_4$ contains a linear subspace $\Lambda $ of dimension $h\geq \max \{2,\dim (\Lambda \cap \operatorname {\mathrm {Sing}}(X_4))+2\}$ or has double points along a linear subspace of dimension $h\geq 3$ , a smooth k-rational point and is otherwise general, then $X_4$ is unirational over k. This improves previous results by A. Predonzan and J. Harris, B. Mazur and R. Pandharipande for quartics. We also provide a density result for the k-rational points of quartic $3$ -folds with a double plane over a number field, and several unirationality results for quintic hypersurfaces over a $C_r$ field.

1 Introduction

An n-dimensional variety X over a field k is rational if it is birational to $\mathbb {P}^n_{k}$ , while X is unirational if there is a dominant rational map . If k is infinite and X is unirational, then the set $X(k)$ of the k-rational points of X is Zariski dense in X.

Since the first half of the twentieth century, the problem of establishing whether a degree d hypersurface $X_d\subset \mathbb {P}^{n+1}$ is rational or unirational has been central in birational projective geometry [Mor42], [Pre49], [Mor52], [IM71], [CG72], [Cil80], [Kol95], [Shi95], [HMP98], [HT00], [dF13], [BRS19], [RS19].

Quadric hypersurfaces with a smooth point are rational and as proven by J. Kollár, building on techniques of B. Segre [Seg43] and Y. I. Manin [Man86, Chapter 2, Section 12], cubic hypersurfaces with a smooth point are unirational [Kol02]. U. Morin proved that a general complex hypersurface $X_d\subset \mathbb {P}^{n+1}$ is unirational provided that n is large enough with respect to d [Mor42]. This result has then been reproved, in a different way, by C. Ciliberto [Cil80] and extended to complete intersections by A. Predonzan [Pre49], K. Paranjape, V. Srinivas [PS92] and L. Ramero [Ram90].

Furthermore, J. Harris, B. Mazur and R. Pandharipande proved that $X_d\subset \mathbb {P}^{n+1}$ is unirational if the codimension of its singular locus is sufficiently big with respect to n and d [HMP98].

Before stating our main results on the unirationality of quartics we briefly survey the state of the art. By the work of U. Morin, a general complex quartic $X_4\subset \mathbb {P}^{n+1}$ with $n\geq 5$ is unirational [Mor36], [Mor52]. V. A. Iskovskikh and Y. I. Manin proved that the group of the birational automorphisms of a smooth quartic $X_4\subset \mathbb {P}^4$ is finite so that $X_4$ is not rational [IM71].

Moreover, J. Harris and Y. Tschinkel showed that if $n\geq 3$ and k is a number field, then for some finite extension $k'$ of k the set of $k'$ -rational points of a smooth quartic $X_4\subset \mathbb {P}^{n+1}$ is dense in the Zariski topology; in other terms, the k-rational points of $X_4$ are potentially dense [HT00]. Despite this great amount of effort, the unirationality of the general quartic $X_4\subset \mathbb {P}^n$ for $n = 4,5$ is still an open problem and only special families of quartic $3$ -folds, called quartics with separable asymptotics, are known to be unirational [Seg60]. For a nice survey on rationality and unirationality problems with a focus on their relation with the notion of rational connection, we refer to A. Verra’s paper [Ver08]. We recall that a projective variety is rationally connected if any two of its points can be joined by a rational curve and refer to C. Araujo’s paper [Ara05] for a survey on the subject.

In this paper, we address the unirationality of quartics $X_4\subset \mathbb {P}^{n+1}$ containing a linear subspace whose dimension is larger than that of the singular locus of $X_4$ or containing a linear subspace with multiplicity two. Our main results in Theorems 3.15 and 3.22 can be summarized as follows:

Theorem 1.1. Let $X_4\subset \mathbb {P}^{n+1}$ be a quartic hypersurface and $\Lambda \subset \mathbb {P}^{n+1}$ an h-plane. Assume that either

1. (i) $n\geq 3, h\geq 2$ , $\dim (\Lambda \cap \operatorname {\mathrm {Sing}}(X_4))\leq h-2$ , $X_4$ contains $\Lambda $ and is not a cone over a smaller-dimensional quartic, or

2. (ii) $n\geq 4, h\geq 3$ , $X_4$ has double points along $\Lambda $ , a point $p\in X_4\setminus \Lambda $ and is otherwise general,

then $X_4$ is unirational.

All along the paper with the word ‘general’ we mean ‘for a nonempty Zariski open subset of the parameter space of the objects we are considering’. We would like to stress that since all the proofs presented in the paper are constructive it is possible, given the equation cutting out the hypersurface, to establish whether or not it is general in the required sense.

Furthermore, for quartic $3$ -folds over a number field we prove, in Proposition 3.14, the following density result.

Theorem 1.2. Let $X_4\subset \mathbb {P}^{4}$ be a quartic hypersurface, over a number field k, having double points along a codimension two linear subspace $\Lambda \subset \mathbb {P}^{4}$ , with a point $p\in X_4\setminus \Lambda $ and otherwise general. The set $X_4(k)$ of the k-rational points of $X_4$ is Zariski dense in $X_4$ .

As Remark 3.17 shows, the assumption on the existence of a point $p\in X_4\setminus \Lambda $ in the case of quartics singular along a linear subspace cannot be dropped. Under extra assumptions on the base field or on the existence of rational points in special subloci of $X_4$ , Theorem 1.1 can be extended to smaller-dimensional quartics. For instance, by Proposition 3.19 a quartic surface $X_4\subset \mathbb {P}^3$ with double points along a line $\Lambda \subset \mathbb {P}^3$ , a point $p\in X_4\setminus \Lambda $ , a double point $q\in X_4\setminus \Lambda $ with $q\neq p$ and otherwise general is unirational. Furthermore, by the second part of Theorem 3.15 a quartic $X_4\subset \mathbb {P}^{4}$ over a $C_r$ field, for which definition we refer to Remark 2.10, with $r = 0,1$ , having double points along a linear subspace $\Lambda $ with $\dim (\Lambda ) = 1,2$ , and otherwise general is unirational. Therefore, in order to complete Theorem 1.1 for all n and h we are left with the following open question.

Question 1.3. Let $X_4\subset \mathbb {P}^{n+1}$ be a quartic hypersurface over a field k such that either

1. (i) $n = 3$ , $X_4$ contains a line, or

2. (ii) $n = 3$ , $X_4$ has double points along a linear subspace $\Lambda $ with $\dim (\Lambda ) = 1,2$ , a point $p\in X_4\setminus \Lambda $ , or

3. (iii) $n = 2$ , $X_4$ has double points along a line $\Lambda $ , a point $p\in X_4\setminus \Lambda $ ,

and $X_4$ is otherwise general. Is then $X_4$ unirational?

Note that a smooth quartic surface $X_4\subset \mathbb {P}^3$ is $K3$ , and hence, it cannot be unirational. As we said, $\text {(ii)}$ has a positive answer when the base field is $C_r$ with $r = 0,1$ , while $\text {(i)}$ is open even over the complex numbers. Since any complex quartic $3$ -fold contains a line, $\text {(i)}$ actually asks about the unirationality of a general quartic $3$ -fold and is probably one of the most interesting unirationality open problems. Since a quartic surface $X_4\subset \mathbb {P}^3$ with a double line is birational to a conic bundle, $\text {(iii)}$ is interesting only when the base field is not algebraically closed.

Note that by considering the generic fiber of the resolution of the linear projection from $\Lambda $ as in the proof of Theorem 3.15 a positive answer to Question 1.3 would extend the first part of Theorem 1.1 to quartic hypersurfaces $X_4\subset \mathbb {P}^{n+1}$ with $n\geq 3$ containing a line and the second part of Theorem 1.1 to quartic hypersurfaces $X_4\subset \mathbb {P}^{n+1}$ with $n\geq 3$ having double points along either a line or a plane and an additional smooth point.

Remark 1.4. The main available results in the spirit of Theorem 1.1 can be found in [Pre49] and [HMP98]. By [Pre49, Theorem 1] a quartic $X_4\subset \mathbb {P}^{n+1}$ containing an h-plane $\Lambda $ is unirational provided that $\operatorname {\mathrm {Sing}}(X_4)\cap \Lambda = \emptyset $ and $h\geq 4$ . The same result has been proved in [HMP98, Corollary 3.7] for $h\geq 97$ .

We would like to stress that both [Pre49] and [HMP98] as well as [Ram90] provide unirationality results for hypersurfaces of arbitrary degree and general unirationality bounds when the base field is algebraically closed.

In the case of quartics, Theorem 1.1 (i) improves [Pre49, Theorem 1] and [HMP98, Corollary 3.7] in two directions: on one side, it is enough to have that $h\geq 2$ , on the other side, $\Lambda $ is allowed to intersect the singular locus of $X_4$ as long as such intersection has codimension at least two in $\Lambda $ .

In the last section, we investigate the unirationality of quintic hypersurfaces and divisors of bidegree $(3,2)$ in products of projective spaces. As a by-product, we get new examples of unirational but not stably rational varieties.

A variety X is stably rational if $X\times \mathbb {P}^m$ is rational for some $m\geq 0$ . Hence, a rational variety is stably rational, and a stably rational variety is unirational. The first examples of stably rational nonrational varieties had been given in [BCTSSD85], where the authors, using Châtelet surfaces, constructed a complex nonrational conic bundle T such that $T\times \mathbb {P}^3$ is rational.

In the last decade, important advances on stable rationality have been made, especially for hypersurfaces in projective spaces [Voi15], [CTP16], [Tot16], [HKT16], [AO18], [Sch18], [BvB18], [HPT18], [Sch19a], [Sch19b], [HPT19]. In [CTP16, Theorem 1.17], J. L. Colliot-Thélène and A. Pirutka proved that a very general smooth complex quartic $3$ -fold is not stably rational. In [Sch19b, Corollary 1.4], S. Schreieder gave the first examples of unirational nonstably rational smooth hypersurfaces. A. Auel, C. Böhning and A. Pirutka proved that a very general divisor of bidegree $(3,2)$ in $\mathbb {P}^3\times \mathbb {P}^2$ , over the complex numbers, is not stably rational. By Theorem 4.7, we get that such a very general divisor is unirational but not stably rational.

Furthermore, thanks to our unirationality results for divisors of bidegree $(3,2)$ we get new results on the unirationality of quintic hypersurfaces over $C_r$ fields and number fields. The literature on the unirationality of quintics is much less rich than that on quartics. A general quintic hypersurface $X_5\subset \mathbb {P}^{n+1}$ , over an algebraically closed field, is unirational if $n\geq 17$ [Mor38]. Furthermore, a quintic $X_5\subset \mathbb {P}^{n+1}$ containing a $3$ -plane and otherwise general is unirational if $n\geq 6$ [CMM08]. To the best of our knowledge, these are the only results on the unirationality of quintics.

Conventions on the base field, terminology and organization of the paper

All along the paper, the base field k will be of characteristic zero. Let X be a variety over k. When we say that X is rational or unirational, without specifying over which field, we will always mean that X is rational or unirational over k. Similarly, we will say that X has a point or contains a variety with certain properties meaning that X has a k-rational point or contains a variety defined over k with the required properties.

In Section 2, we will introduce the notation, prove some preliminary results about the relation between hypersurfaces in projective spaces and certain divisors in projective bundles and give an immediate generalization of a unirationality criterion due to F. Enriques. In Section 3, we will investigate the unirationality of quartic hypersurfaces and cubic complexes that are complete intersections of a quadric and a cubic. Finally, in Section 4 we will address the unirationality of quintics and divisors in products of projective spaces.

2 Hypersurfaces and divisors in projective bundles

Let $a_0,\dots ,a_{h+1} \in \mathbb {Z}_{\geq 0}$ be nonnegative integers, and consider the simplicial toric variety $\mathcal {T}_{a_0,\dots ,a_{h+1}}$ with Cox ring

$$ \begin{align*}\operatorname{\mathrm{Cox}}(\mathcal{T}_{a_0,\dots,a_{h+1}})\cong k[x_0,\dots,x_{n-h},y_0,\dots,y_{h+1}]\end{align*} $$

$\mathbb {Z}^2$ -grading given, with respect to a fixed basis $(H_1,H_2)$ of $\operatorname {\mathrm {Pic}}(\mathcal {T}_{a_0,\dots ,a_{h+1}})$ , by the following matrix

$$ \begin{align*}\left(\begin{array}{cccccc} x_0 & \dots & x_{n-h} & y_0 & \dots & y_{h+1}\\ \hline 1 & \dots & 1 & -a_0 & \dots & -a_{h+1}\\ 0 & \dots & 0 & 1 & \dots & 1 \end{array}\right) \end{align*} $$

and irrelevant ideal $(x_0,\dots ,x_{n-h})\cap (y_0,\dots ,y_{h+1})$ . Then

$$ \begin{align*}\mathcal{T}_{a_0,\dots,a_{h+1}}\cong \mathbb{P}(\mathcal{E}_{a_0,\dots,a_{h+1}})\end{align*} $$

with $\mathcal {E}_{a_0,\dots ,a_{h+1}}\cong \mathcal {O}_{\mathbb {P}^{n-h}}(a_0)\oplus \dots \oplus \mathcal {O}_{\mathbb {P}^{n-h}}(a_{h+1})$ . The secondary fan of $\mathcal {T}_{a_0,\dots ,a_{h+1}}$ is as follows

where $H_1 = (1,0)$ corresponds to the sections $x_0,\dots ,x_{n-h}$ , $H_2 = (0,1)$ , and $v_i = (-a_i,1)$ corresponds to the section $y_i$ for $i = 0,\dots ,h+1$ .

Definition 2.1. A divisor $D\subset \mathcal {T}_{a_0,\dots ,a_{h+1}}$ of multidegree $(\delta _{d,0,\dots ,0},\dots ,\delta _{0,\dots ,0,d}; d)$ is a hypersurface given by an equation of the following form

(2.2) $$ \begin{align} D := \left\lbrace\sum_{0\leq i_0\leq \dots \leq i_{h+1}\: |\: i_0+ \dots + i_{h+1}=d}\sigma_{i_0,\dots,i_{h+1}}(x_0,\dots,x_{n-h})y_0^{i_0}\dots y_{h+1}^{i_{h+1}} = 0\right\rbrace\subset \mathcal{T}_{a_0,\dots,a_{h+1}}, \end{align} $$

where $\sigma _{i_0,\dots ,i_{h+1}}\in k[x_0,\dots ,x_{n-h}]_{\delta _{i_0,\dots ,i_{h+1}}}$ and

(2.3) $$ \begin{align} \delta_{d,0,\dots,0}-da_0 = \delta_{d-1,1,0,\dots,0}-(d-1)a_0-a_1 = \dots = \delta_{0,\dots,0,d}-da_{h+1}. \end{align} $$

Without loss of generality, we may assume that $a_0\geq a_1\geq \dots \geq a_{h+1}$ so that (2.3) yields $\delta _{d,0,\dots ,0} \geq \delta _{0,d,0,\dots ,0}\geq \dots \geq \delta _{0,\dots ,0,d}$ .

Lemma 2.4. Let $X_d\subset \mathbb {P}^{n+1}$ be a hypersurface of degree d having multiplicity m along an h-plane $\Lambda $ , and $\widetilde {X}_d$ the blowup of $X_d$ along $\Lambda $ with exceptional divisor $\widetilde {E}\subset \widetilde {X}_d$ . Then $\widetilde {X}_d$ is isomorphic to a divisor of multidegree

$$ \begin{align*}(d,d-1,\dots,d-1,\dots,d-j,\dots,d-j,\dots,m,\dots,m; d-m)\end{align*} $$

in $\mathcal {T}_{1,0,\dots ,0}$ , where $d-j$ is repeated $\binom {h+j}{j}$ times for $j = 0,\dots ,d-m$ . The exceptional divisor $\widetilde {E}$ is a divisor of bidegree $(m,d-m)$ in $\mathbb {P}^{n-h}\times \mathbb {P}^{h}$ .

Proof. We may assume that $\Lambda = \{z_0 = \dots = z_{n-h} = 0\}$ and

$$ \begin{align*}X_d = \left\lbrace \sum_{m_0+\dots+m_{n-h} = m}z_0^{m_0}\dots z_{n-h}^{m_{n-h}}A_{m_0,\dots,m_{n-h}}(z_0,\dots,z_{n+1}) = 0\right\rbrace\subset\mathbb{P}^{n+1}\end{align*} $$

with $A_{m_0,\dots ,m_{n-h}}\in k[z_0,\dots ,z_{n+1}]_{d-m}$ . The blowup of $\mathbb {P}^{n+1}$ along $\Lambda $ is the simplicial toric variety $\mathcal {T}$ with Cox ring

$$ \begin{align*}\operatorname{\mathrm{Cox}}(\mathcal{T})\cong k[x_0,\dots,x_{n-h},y_0,\dots,y_{h+1}]\end{align*} $$

$\mathbb {Z}^2$ -grading given, with respect to a fixed basis $(H_1,H_2)$ of $\operatorname {\mathrm {Pic}}(\mathcal {T})$ , by the following matrix:

$$ \begin{align*}\left(\begin{array}{cccccccc} x_0 & \dots & x_{n-h} & y_0 & y_1 & \dots & y_{h+1}\\ \hline 1 & \dots & 1 & 0 & 1 & \dots & 1\\ -1 & \dots & -1 & 1 & 0 & \dots & 0 \end{array}\right) \end{align*} $$

and irrelevant ideal $(x_0,\dots ,x_{n-h})\cap (y_0,\dots ,y_{h+1})$ . Substituting in the above matrix the first row with the sum of the rows and then swapping the rows an multiplying the top row by $-1$ we get to the following grading matrix

(2.5) $$ \begin{align} \left(\begin{array}{cccccccc} x_0 & \dots & x_{n-h} & y_0 & y_1 & \dots & y_{h+1}\\ \hline 1 & \dots & 1 & -1 & 0 & \dots & 0\\ 0 & \dots & 0 & 1 & 1 & \dots & 1 \end{array}\right) \end{align} $$

and hence $\mathcal {T} = \mathcal {T}_{1,0,\dots ,0}$ . The blow-down morphism is given by

$$ \begin{align*}\begin{array}{cccc} \phi: & \mathcal{T}_{1,0,\dots,0} & \longrightarrow & \mathbb{P}^{n+1}\\ & (x_0,\dots,x_{n-h},y_0,\dots,y_{h+1}) & \mapsto & [x_0y_0:\dots :x_{n-h}y_0:y_1:\dots :y_{h+1}] \end{array} \end{align*} $$

and the exceptional divisor is $E = \{y_0 = 0\}$ . Hence, the strict transform of $X_d$ is defined by

(2.6) $$ \begin{align} \widetilde{X}_d = \left\lbrace \sum_{m_0+\dots+m_{n-h} = m}x_0^{m_0}\dots x_{n-h}^{m_{n-h}}A_{m_0,\dots,m_{n-h}}(x_0y_0,\dots,x_{n-h}y_0,y_1,\dots,y_{h+1}) = 0\right\rbrace\subset\mathcal{T}_{1,0,\dots,0}, \end{align} $$

and the claim on the multidegree follows. Note that (2.5) yields that $E \cong \mathbb {P}^{n-h}_{(x_0,\dots ,x_{n-h})}\times \mathbb {P}^h_{(y_1,\dots ,y_{h+1})}$ and hence $\widetilde {E} = \widetilde {X}_d\cap E \subset \mathbb {P}^{n-h}_{(x_0,\dots ,x_{n-h})}\times \mathbb {P}^h_{(y_1,\dots ,y_{h+1})}$ is a divisor of bidegree $(m,d-m)$ .

The following is a straightforward generalization of a unirationality criterion for conic bundles due to F. Enriques [IP99, Proposition 10.1.1].

Proposition 2.7. Let $f:X\rightarrow Y$ be a fibration over a unirational variety Y with X an irreducible variety. Assume that there exists a unirational subvariety $Z\subset X$ such that $f_{|Z}:Z\rightarrow Y$ is dominant, consider the fiber product

and denote by $X_{Z,\eta }$ the generic fiber of $\widetilde {f}:X_Z\rightarrow Z$ . Assume that $X_{Z,\eta }$ is integral. Finally, assume that $X_{Z,\eta }$ is unirational over $k(Z)$ if and only if it has a $k(Z)$ -rational point. Then X is unirational.

Proof. Since $X_{Z,\eta }$ is integral after replacing Z by an open subset, we may assume that $X_Z$ is irreducible.

Now, note that the dominant morphism $f_{|Z}:Z\rightarrow Y$ yields a rational section of $\widetilde {f}:X_Z\rightarrow Z$ . So $X_{Z,\eta }$ has a $k(Z)$ -rational point, and hence, it is unirational over $k(Z)$ . Therefore, $X_Z$ is unirational, and hence, X is unirational as well.

We will apply Proposition 2.7 to fibrations in irreducible m-dimensional quadric or cubic hypersurfaces. In these cases, $X_{Z,\eta }\subset \mathbb {P}^{m+1}_{k(Z)}$ is an irreducible quadric or cubic hypersurface over $k(Z)$ . So after replacing Z by an open subset, we may assume that $X_Z$ is irreducible.

Remark 2.8. A quadric hypersurface over a field k with a smooth point is rational. Furthermore, by [Kol02, Theorem 1.2] a cubic hypersurface of dimension at least two with a point and which is not a cone is unirational.

Definition 2.9. Fix a real number $r\in \mathbb {R}_{\geq 0}$ . A field k is $C_r$ if and only if every homogeneous polynomial $f\in k[x_0,\dots ,n_n]_d$ of degree $d> 0$ in $n+1$ variables with $n+1> d^r$ has a nontrivial zero in $k^{n+1}$ .

Remark 2.10. (Lang’s theorem) If k is a $C_r$ field, $f_1,\dots , f_s\in k[x_0,\dots ,n_n]_d$ are homogeneous polynomials of the same degree and $n+1> sd^r$ , then $f_1,\dots ,f_s$ have a nontrivial common zero in $k^{n+1}$ [Poo17, Proposition 1.2.6]. Furthermore, if k is $C_r$ , then $k(t)$ is $C_{r+1}$ [Poo17, Theorem 1.2.7].

In the last section, we will need the following.

Proposition 2.11. Let $D\subset \mathcal {T}_{a_0,\dots ,a_{h+1}}\rightarrow \mathbb {P}^{n-h}$ be a divisor of multidegree $(\delta _{2,0,\dots ,0},\dots ,\delta _{0,\dots ,0,2},2)$ . Then D is birational to a hypersurface $X_{\delta _{2,0,\dots ,0}+2}^n\subset \mathbb {P}^{n+1}$ of degree $\delta _{2,0,\dots ,0}+2$ having multiplicity $\delta _{2,0,\dots ,0}$ along an h-plane $\Lambda $ and multiplicity two along an $(n-h-1)$ -plane $\Lambda '$ such that $\Lambda \cap \Lambda ' = \emptyset $ .

Proof. Write $D\subset \mathcal {T}_{a_0,\dots ,a_{h+1}}$ as in (2.2) and dehomogenize with respect to $x_{n-h}$ and $y_{h+1}$ to get an affine hypersurface

$$ \begin{align*}X = \{f(x_0,\dots,x_{n-h-1},y_0,\dots,y_h) = 0\}\subset\mathbb{A}^{n+1}_{(x_0,\dots,x_{n-h-1},y_0,\dots,y_h)}. \end{align*} $$

Now, we introduce a new variable that we will keep denoting by $x_{n-h}$ and homogenize f in order to get a polynomial $\overline {f}(x_0,\dots ,x_{n-h-1},y_0,\dots ,y_h,x_{n-h})$ which is homogeneous of degree $\delta _{2,0,\dots ,0}+2$ . Note that $\overline {f}$ has the following form

(2.12) $$ \begin{align} \overline{f} = \sum_{i_0+\dots +i_{n-h} = \delta_{2,0,\dots,0}}x_0^{i_0}\dots x_{n-h}^{i_{n-h}}A_{i_0,\dots,i_{n-h}}(y_0,\dots,y_h,x_{n-h}) \end{align} $$

with $A_{i_0,\dots ,i_{n-h}}\in k[y_0,\dots ,y_h,x_{n-h}]_2$ for all $0\leq i_0\leq \dots \leq i_{n-h}\leq \delta _{2,0,\dots ,0}$ . The hypersurface

$$ \begin{align*}X_{\delta_{2,0,\dots,0}+2}^n = \{\overline{f}(x_0,\dots,x_{n-h-1},y_0,\dots,y_h,x_{n-h}) = 0\}\subset\mathbb{P}^{n+1} \end{align*} $$

is birational to X, and hence, it is birational to D as well. To conclude, set

$$ \begin{align*}\Lambda = \{x_0 = \dots = x_{n-h} = 0\},\: \Lambda' = \{y_0 = \dots = y_{h} = x_{n-h} = 0\} \end{align*} $$

and note that (2.12) yields $\operatorname {\mathrm {mult}}_{\Lambda }X_{\delta _{2,0,\dots ,0}+2}^n = \delta _{2,0,\dots ,0}$ and $\operatorname {\mathrm {mult}}_{\Lambda '}X_{\delta _{2,0,\dots ,0}+2}^n = 2$ .

3 Cubic complexes and Quartics

In this section, we investigate the unirationality of quartic hypersurfaces containing linear subspaces and quadrics, and of complete intersections of quadric and cubic hypersurfaces. The following is an immediate consequence of [CMM07, Theorem 2.1].

Lemma 3.1. Let $Y_{2,3} = Y_2 \cap Y_3\subset \mathbb {P}^{n+2}$ with $n\geq 3$ be a smooth complete intersection, of a smooth quadric $Y_2$ and a cubic $Y_3$ , defined over a field k with $\operatorname {\mathrm {char}}(k)\neq 2,3$ . If $Y_{2}$ contains a $2$ -plane $\Pi $ , then $Y_{2,3}$ is unirational.

Proof. For $n = 3$ , the statement has been proven in [CMM07, Theorem 2.1]. Now, consider the incidence varieties

$$ \begin{align*}\begin{array}{ll} W_2 & = \{(y,H) \: | \: y\in H\cap Y_{2}\}\subset Y_{2}\times \mathbb{G}(n-4,n-1); \\ W_3 & = \{(y,H) \: | \: y\in H\cap Y_{3}\}\subset Y_{3}\times \mathbb{G}(n-4,n-1); \\ W_{2,3} & = \{(y,H) \: | \: y\in H\cap Y_{2,3}\} = W_2\cap W_3\subset Y_{2,3}\times \mathbb{G}(n-4,n-1), \end{array} \end{align*} $$

where $\mathbb {G}(n-4,n-1)$ is the Grassmannian of $5$ -planes in $\mathbb {P}^{n+2}$ containing $\Pi $ . Let $W_{2,\eta }, W_{3,\eta }, W_{2,3,\eta }$ be the generic fibers of the second projection onto $\mathbb {G}(n-4,n-1)$ from $W_2,W_3,W_{2,3}$ , respectively.

The $2$ -plane $\Pi \subset Y_{2}$ yields a $2$ -plane $\Pi '\subset W_{2,\eta }$ defined over $k(t_1,\dots ,t_{3(n-3)})$ , and since $Y_2$ is smooth, we have that $W_{2,\eta }$ is smooth.

Hence, $W_{2,3,\eta }\subset \mathbb {P}^5_{k(t_1,\dots ,t_{3(n-3)})}$ is a smooth complete intersection over $k(t_1,\dots ,t_{3(n-3)})$ of a quadric and a cubic satisfying the hypotheses of [CMM07, Theorem 2.1], and so $W_{2,3,\eta }$ is unirational over the function field $k(t_1,\dots ,t_{3(n-3)})$ . As in the proof of Proposition 2.7, after replacing $\mathbb {G}(n-4,n-1)$ by an open subset we may assume that $W_{2,3}$ is irreducible.

Therefore, $W_{2,3}$ is unirational over the base field k, and to conclude it is enough to observe that the first projection $W_{2,3}\rightarrow Y_{2,3}$ is dominant.

Lemma 3.2. Let $Q_0\subset \mathbb {P}^{n+1}$ be a smooth $(n-1)$ -dimensional quadric and $X_d\subset \mathbb {P}^{n+1}$ an irreducible hypersurface of degree d containing $Q_0$ with multiplicity one and otherwise general. Then there exists a complete intersection

$$ \begin{align*}Y_{2,d-1} = Y_2\cap Y_{d-1}\subset\mathbb{P}^{n+2}\end{align*} $$

of a quadric $Y_2$ and a hypersurface $Y_{d-1}$ of degree $d-1$ such that:

1. (i) $Y_{2,d-1}$ has a point $v\in Y_{2,d-1}$ of multiplicity $d-2$ , and

2. (ii) the linear projection from v yields a birational map .

Furthermore, $Y_2$ is smooth and $\operatorname {\mathrm {mult}}_{v}Y_{d-1} = d-2$ .

Proof. We may write $Q_0 = \{z_1 = Q = 0\}$ with $Q\in k[z_0,\dots ,z_{n+1}]_2$ . Then $X_d$ is of the form

$$ \begin{align*}X_d = \{z_1A+BQ = 0\}\end{align*} $$

with $A\in k[z_0,\dots ,z_{n+1}]_{d-1}$ and $B\in k[z_0,\dots ,z_{n+1}]_{d-2}$ . Consider the quadric $Y_2 = \{z_1u-Q = 0\}$ with $z_0,\dots ,z_{n+1},u$ homogeneous coordinates on $\mathbb {P}^{n+2}$ . Since $Q_0$ is smooth, we may assume that $Q = z_0^2+\dots + z_{n+1}^2$ . Hence, $Y_2$ is a smooth as well.

Denote by $CX_d$ , $CQ_0$ the cones, respectively, over $X_d$ and $Q_0$ with vertex $v = [0:\dots :0:1]$ . Then $CQ_0\subset Y_2$ and

$$ \begin{align*}Y_2\cap CX_d = CQ_0 \cup \{z_1u-Q = A+uB = 0\}.\end{align*} $$

Set $Y_{d-1} = \{A+uB = 0\}$ and $Y_{2,d-1} = \{z_1u-Q = A+uB = 0\}$ . Note that the tangent space of $Y_2$ at v is the hyperplane $\{z_1 = 0\}$ , the tangent cone of $Y_{d-1}$ at v is given by $\{B = 0\}$ and the tangent cone of $Y_{2,d-1}$ at v is cut out by $\{z_1 = B = 0\}$ . Hence,

$$ \begin{align*}\operatorname{\mathrm{mult}}_{v}Y_{d-1} = \operatorname{\mathrm{mult}}_{v}Y_{2,d-1} = d-2.\end{align*} $$

Therefore, if $p\in Y_{2,d-1}$ is a general point the line $\left \langle v,p\right \rangle $ intersects $Y_{2,d-1}$ just in v with multiplicity $d-2$ and in p with multiplicity one. So the projection is birational onto $\overline {\pi _{v}(Y_{2,d-1})}$ which must then be a hypersurface of degree $2(d-1)-(d-2) = d$ . To conclude, it is enough to note that since $Y_{2,d-1}$ is not a cone of vertex v and $Y_{2,d-1}\subset CX_d$ we have $\overline {\pi _{v}(Y_{2,d-1})} = X_d$ .

Lemma 3.3. Consider the complete intersection

$$ \begin{align*}Y_{2,d-1} = Y_2\cap Y_{d-1}\subset\mathbb{P}^{n+2},\end{align*} $$

where $Y_2 = \{z_1u-A_1 = 0\}\subset \mathbb {P}^{n+2}_{(z_0,\dots ,z_{n-1},u)}$ and

$$ \begin{align*}Y_{d-1} = \{z_0^{d-2}u+z_0^{d-3}A_2+z_0^{d-4}z_1A_3+\dots +z_1^{d-3}A_{d-1} = 0\}\subset\mathbb{P}^{n+2}_{(z_0,\dots,z_{n-1},u)}\end{align*} $$

with $A_i$ general homogeneous polynomial of degree two. Assume that the $(n-2)$ -dimensional quadric $\overline {Q} = \{z_0 = z_1 = A_1 = 0\}$ has a point.

Then there exists a variety W, with a surjective morphism onto a rational surface S, such that if W is unirational, then there is a dominant rational map and hence $Y_{2,d-1}$ is unirational as well. Furthermore, the general fiber of $W\rightarrow S$ is a complete intersection of a quadric and a hypersurface of degree $d-3$ .

Proof. Set $v = [0:\dots :0:1]$ and $H = \{z_0 = z_1 = 0\}\subset \mathbb {P}^{n+2}$ . Then

$$ \begin{align*}\operatorname{\mathrm{mult}}_{v}Y_{d-1} = d-2,\: \operatorname{\mathrm{mult}}_{H}Y_{d-1} = d-3.\end{align*} $$

Take a general point $p\in H$ . The lines through p that intersect $Y_{d-1}$ at p with multiplicity at least $d-2$ are parametrized by a hypersurface $W_{d-3}$ cut out in the $(n+1)$ -dimensional projective space $\mathbb {P}(T_p\mathbb {P}^{n+2})$ of lines through p by a polynomial in $k[z_0,z_1]_{d-3}$ .

Now, consider the cone $C\overline {Q}$ over the $(n-2)$ -dimensional quadric $\overline {Q} = \{z_0 = z_1 = A_1 = 0\}$ , and take a general point $p \in C\overline {Q}$ . Note that $C\overline {Q}\subset Y_2$ . The lines through p that are contained in $Y_2$ are parametrized by a quadric hypersurface $W_2\subset \mathbb {P}(T_pY_2)$ .

Let $\mathcal {F} = \mathcal {T}_{Y_2|C\overline {Q}}$ be the restriction of the tangent sheaf of $Y_2$ to $C\overline {Q}$ . Summing up, there is a subvariety $W_{2,d-3}\subset \mathbb {P}(\mathcal {F})$ with a surjective morphism $\rho :W_{2,d-3}\rightarrow C\overline {Q}$ whose fiber over a general point of $p\in C\overline {Q}$ is a complete intersection of a quadric and a hypersurface of degree $d-3$ in $\mathbb {P}^{n}$ . Hence, $\dim (W_{2,d-3}) = 2n-3$ .

By hypothesis, $\overline {Q}$ has a point. Let $\overline {C}$ be a conic in $\overline {Q}$ through this point and S the cone over $\overline {C}$ with vertex v. Then $S\subset C\overline {Q}$ is a rational surface. Set $W = \rho ^{-1}(S)$ and $W_s = \rho ^{-1}(s)$ for $s\in S$ . A general point $w\in W$ represents a pair $(s,l_s)$ , where $s\in S$ and $l_s$ is a line through s which is contained in $Y_2$ and intersects $Y_{d-1}$ with multiplicity $d-2$ at s. Since $\deg (Y_{d-1}) = d-1$ the line $l_s$ intersects $Y_{d-1}$ just at one more point $x_{(s,l_s)}\in Y_2 \cap Y_{d-1} = Y_{2,d-1}$ and we get a rational map

If W is unirational, in order to prove that $\psi $ is dominant it is enough to prove that the induced map

between the algebraic closures is dominant. Take a general point $p \in \overline {Y}_{2,d-1}$ , and assume that $x_{(s,l_s)} = p$ . Then $l_s$ lies in the tangent space of $Y_2$ at p which is given by $\{L = 0\}$ . Such tangent space intersects $\overline {S}$ in a conic, and further intersecting with $W_{d-3}$ we see that there are finitely may points $s\in \overline {S}$ such that $x_{(s,l_s)} = p$ for some $l_s\in W_s$ . Furthermore, if $x_{(s,l_s)} = x_{(s,l_s')}$ , then $l_s = l_s'$ . Hence, $\overline {\psi }$ is generically finite and since $\dim (W) = \dim (Y_{2,d-1})$ we conclude that $\overline {\psi }$ is dominant.

Proposition 3.4. Let $X_d\subset \mathbb {P}^{n+1}$ be a hypersurface of degree d having multiplicity $d-2$ along an $(n-1)$ -plane $\Lambda \subset \mathbb {P}^{n+1}$ . Assume that there is a quadric $Q_p$ in the quadric fibration induced by the projection from $\Lambda $ such that the quadric $Q_p\cap \Lambda $ is smooth and has a point and that $X_d$ is otherwise general.

Then there exist a rational surface S and a variety W with a morphism onto S such that if W is unirational, then there is a dominant rational map and hence $X_d$ is unirational as well.

Proof. We may write $\Lambda = \{z_0 = z_1 = 0\}$ and

$$ \begin{align*}X_d = \{z_0^{d-2}A_1 +z_{0}^{d-3}z_1A_2+\dots +z_1^{d-2}A_{d-1} = 0\}\end{align*} $$

with $A_i \in k[z_0,\dots ,z_{n+1}]_2$ . Note that $X_d$ contains the smooth $(n-1)$ -dimensional quadric $Q_p = \{z_1 = A_1 = 0\}$ . Hence, by Lemma 3.2 there is an irreducible complete intersection $Y_{2,d-1}\subset \mathbb {P}^{n+2}$ which is birational to $X_d$ .

Consider the quadric $Y_2 = \{z_1u-A_1 = 0\}\subset \mathbb {P}^{n+2}$ with homogeneous coordinates $z_0,\dots ,z_{n+1},u$ , and let $CX_d$ be the cone over $X_d$ as in the proof of Lemma 3.2. The intersection $Y_2\cap CX_d$ has two components: the cone $CQ_p$ over $Q_p$ and the degree $d-1$ hypersurface

$$ \begin{align*}Y_{d-1} = \{z_0^{d-2}u+z_0^{d-3}A_2+z_0^{d-4}z_1A_3+\dots +z_1^{d-3}A_{d-1} = 0\}.\end{align*} $$

Let $W\rightarrow S$ be the fibration constructed in Lemma 3.3 starting from the complete intersection $Y_{2,d-1} = Y_2\cap Y_{d-1}$ . If W is unirational, Lemma 3.3 yields a dominant rational map

. Finally, let

be the dominant rational map in Lemma 3.2. By considering the composition

we get a dominant rational map

, and hence $X_d$ is unirational.

Proposition 3.5. Let $Y_{2,3} = Y_2\cap Y_3\subset \mathbb {P}^{n+2}$ be a complete intersection of a quadric and a cubic of the following form

$$ \begin{align*}Y_{2,3} =\{uz_1 - A = z_0^2u+z_0B+z_1C = 0\} \end{align*} $$

with $A,B,C\in k[z_0,\dots ,z_{n+1}]_2$ general. If the quadric $\overline {Q} = \{z_0 = z_1 = A = 0\}$ is smooth and has a point, then $Y_{2,3}$ is unirational.

Proof. By Lemma 3.3 with $d = 4$ , there exists a variety W with a morphism onto a rational surface S whose general fiber is a complete intersection of a quadric and a hyperplane. Hence, W has a structure of quadric bundle $W\rightarrow S$ over S with $(n-2)$ -dimensional quadrics as fibers.

By the proof of Lemma 3.3, S is a cone over a conic and it is contained in $Y_2 = \{uz_1 - A = 0\}$ . In particular, any line in S through its vertex is contained in $Y_2$ . Moreover, S is contained in the intersection of $Y_3$ with its tangent cone at v. Hence, the lines in S through its vertex yield a section of the quadric bundle $W\rightarrow S$ and so W is rational. Finally, to conclude it is enough to apply Lemma 3.3.

Corollary 3.6. Let $X_4\subset \mathbb {P}^{n+1}$ be a quartic hypersurface having multiplicity two along an $(n-1)$ -plane $\Lambda \subset \mathbb {P}^{n+1}$ with $n\geq 3$ . Assume that there is a quadric $Q_p$ in the quadric fibration induced by the projection from $\Lambda $ such that the quadric $Q_{\Lambda } = Q_p\cap \Lambda $ is smooth and has a point and that $X_4$ is otherwise general. Then $X_4$ is unirational.

Proof. Up to a change of coordinates, we may assume that $\Lambda = \{z_0 = z_1 = 0\}$ , $X_4$ is given by

$$ \begin{align*}X_4 = \{z_0^2A+z_0z_1B+z_1^2C = 0\}\subset\mathbb{P}^{n+1} \end{align*} $$

and $Q_{\Lambda } = \{z_1 = A = 0\}$ . The intersection $\{z_1u-A = z_0^2A+z_0z_1B+z_1^2C = 0\}\subset \mathbb {P}^{n+2}_{(z_0,\dots ,z_{n+1},u)}$ has two components: $\{z_1 = A = 0\}$ and

$$ \begin{align*}Y_{2,3} = \{z_1u-A = uz_0^2+z_0B+z_1^2C = 0\}\subset\mathbb{P}^{n+2}_{(z_0,\dots,z_{n+1},u)}. \end{align*} $$

By Lemma 3.2 and its proof, there exists a birational map . To conclude, it is enough to note that $Y_{2,3}$ is a complete intersection of the form covered by Proposition 3.5.

Corollary 3.7. Let $X_4\subset \mathbb {P}^{n+1}$ , with $n\geq 5$ , be a quartic hypersurface containing an $(n-1)$ -dimensional smooth quadric $Q_0$ which contains a $2$ -plane and otherwise general. Then $X_4$ is unirational.

Proof. By Lemma 3.2, $X_4$ is birational to a complete intersection $Y_{2,3} = Y_2\cap Y_3$ , where $Y_2$ is a smooth quadric. By the proof of Lemma 3.2, we have that since $Q_0$ contains a $2$ -plane $Y_2$ contains a $2$ -plane as well. Hence, $Y_2$ is a smooth quadric containing a $2$ -plane and we conclude by Lemma 3.1.

Corollary 3.8. Let $X_4\subset \mathbb {P}^{n+1}$ , with $n\geq 2$ , be a quartic hypersurface having multiplicity two along an $(n-1)$ -dimensional smooth quadric Q, with a point and otherwise general. Then $X_4$ is unirational.

Proof. Slightly modifying the proof of Lemma 3.2, we see that in this case $X_4$ is birational to a complete intersection $Y_2\cap Y_2'\subset \mathbb {P}^{n+2}$ of two quadrics, and hence, the claim follows from [CTSSD87, Proposition 2.3].

In the following, we will investigate the unirationality of quartic hypersurfaces by constructing explicit birational maps to divisors in products of projective spaces. In particular, we will get an improvement of Corollary 3.6 when $n\geq 4$ .

Lemma 3.9. Consider the hypersurface

$$ \begin{align*}X_d = \{z_0^{d-2}(z_0L+Q) + z_0^{d-3}z_1 A_1+\dots + z_0z_1^{d-3}A_{d-3}+z_1^{d-2}A_{d-2} = 0\}\subset\mathbb{P}^{n+1}_{(z_0,\dots,z_{n+1}),} \end{align*} $$

where $A_i = A_i^2 + z_0A_i^1+z_0^2A_i^0$ , and $L,A_i^1\in k[z_1,\dots ,z_{n+1}]_1$ , $Q,A_i^2\in k[z_1,\dots ,z_{n+1}]_2$ , $A_i^0\in k$ for $i = 1,\dots ,d-2$ . Then $X_d$ is birational to the divisor

$$ \begin{align*}Y_{(d-1,2)} = \left\{\sum_{i=0}^{d-1}x_0^{d-1-i}x_1^iB_i = 0\right\}\subset\mathbb{P}^1_{(x_0,x_1)}\times \mathbb{P}^{n}_{(w_1,\dots, w_{n+1})} \end{align*} $$

of bidegree $(d-1,2)$ , where $B_0 = w_1L+A_1^0w_1^2$ , $B_1 = Q+w_1A_1^1+w_1^2A_2^0$ , $B_j = A_{j-1}^2+w_1A_j^1+w_1^2A_{j+1}^0$ for $j = 2,\dots ,d-3$ , $B_{d-2} = A_{d-3}^2+w_1A_{d-2}^1$ , $B_{d-1} = A_{d-2}^2$ .

Proof. Note that $X_d$ passes through the point $p = [1:0:\dots :0]$ , and the rational map

is birational with birational inverse

where $L = L(w_1,\dots ,w_{n+1})$ . Note that $\varphi ^{-1}$ contracts the divisor $\{L = 0\}$ to the point p. The strict transform of $X_d$ via $\varphi ^{-1}$ is given by

$$ \begin{align*}\widetilde{X}_d = \{w_0^{d-2}(w_0w_1L+LQ) &+ w_0^{d-3}(w_0^2w_1^2A_1^0 + w_0w_1LA_1^1+L^2A_1^2)+ \dots\\ &+L^{d-3}(w_0^2w_1^2A_{d-2}^0 + w_0w_1LA_{d-2}^1+L^2A_{d-2}^2)=0\}, \end{align*} $$

which we rewrite as

$$ \begin{align*}\widetilde{X}_d = \{w_0^{d-1}(w_1L+w_1^2A_1^0)& +w_0^{d-2}L(Q+w_1A_1^1+w_1^2A_2^0)+\dots\\ & +w_0L^{d-2}(A_{d-3}^2+w_1A_{d-2}^1) + L^{d-1}A_{d-2}^2 = 0\}. \end{align*} $$

Finally, substituting $w_0 = \frac {x_0}{x_1}L$ we get the equation cutting out the divisor $Y_{(d-1,2)}\subset \mathbb {P}^1_{(x_0,x_1)}\times \mathbb {P}^{n}_{(w_1:\dots : w_{n+1})}$ in the statement.

Proposition 3.10. Let $Y_{(3,2)}$ be a general divisor of the form

$$ \begin{align*}Y_{(3,2)} = \left\{\sum_{i=0}^{3}x_0^{3-i}x_1^iB_i = 0\right\}\subset\mathbb{P}^1_{(x_0,x_1)}\times \mathbb{P}^{n}_{(w_1,\dots, w_{n+1}),} \end{align*} $$

where $B_0 = w_1L+A_1^0w_1^2$ , $B_1 = Q+w_1A_1^1+w_1^2A_2^0$ , $B_{2} = A_{1}^2+w_1A_2^1$ , $B_{3} = A_2^2$ , and $L,A_i^1\in k[w_1,\dots ,w_{n+1}]_1$ , $Q,A_i^2\in k[w_1,\dots ,w_{n+1}]_2$ , $A_i^0\in k$ for $i = 1,2$ are general. If $n\geq 4$ , then $Y_{(3,2)}$ is unirational.

Proof. Consider the rational map

(3.11)

where $\eta _i = B_ix_0^{2}+B_{i+1}x_0x_1+\dots +B_{3}x_0^{i-1}x_1^{3-i}$ for $i = 1,2,3$ . By [Ott15, Theorem 1.1 (ii)] the rational map

yields a small transformation

, where $Y_{(3,2)}^{+}\subset \mathbb {P}^{2}_{(u_1,u_2,u_{3})}\times \mathbb {P}^{n}_{(w_1,\dots , w_{n+1})}$ is cut out by the minors of order three of the following matrix:

(3.12) $$ \begin{align} M_{(u_1,u_2,u_{3})} = \left( \begin{array}{ccc} 0 & u_1 & B_0\\ -u_1 & u_2 & B_1\\ -u_{2} & u_{3} & B_{2}\\ -u_{3} & 0 & B_{3} \end{array} \right). \end{align} $$

Consider the point $p = ([1:0],[0:\dots :0:1])\in Y_{(3,2)}$ , its image

$$ \begin{align*}q &= ([B_1(0,\dots,0,1),B_{2}(0,\dots,0,1)],B_{3}(0,\dots,0,1)])\\ &= ([Q(0,\dots,0,1),A_{1}^2(0,\dots,0,1),A_{2}^2(0,\dots,0,1)]) \end{align*} $$

via $\eta $ and set $\overline {u}_1 = Q(0,\dots ,0,1),\overline {u}_{2} = A_{1}^2(0,\dots ,0,1),\overline {u}_{3} = A_{2}^2(0,\dots ,0,1)$ . Let $F_{\overline {u}}$ be the fiber of the first projection $\pi _1:\mathbb {P}^{2}_{(u_1,u_2,u_{3})}\times \mathbb {P}^{n}_{(w_1,\dots , w_{n+1})}\rightarrow \mathbb {P}^{2}_{(u_1,u_2,u_{3})}$ over $\overline {u} = [\overline {u}_1:\overline {u}_2:\overline {u}_{3}]$ . Then

$$ \begin{align*}F_{\overline{u}} = \{\operatorname{\mathrm{rank}}(M_{(\overline{u}_1,\overline{u}_2,\overline{u}_{3})}) < 3\}\subset \mathbb{P}^{n}_{(w_1,\dots, w_{n+1})} \end{align*} $$

is a complete intersection of two quadrics. Note that $q\in F_{\overline {u}}$ and since the $\overline {u}_i$ are general, $F_{\overline {u}}$ is smooth. Therefore, if $n\geq 4$ [CTSSD87, Proposition 2.3] yields the unirationality of $F_{\overline {u}}$ . The strict transform of $F_{\overline {u}}$ via $\eta ^{+}$ is given by

$$ \begin{align*}\widetilde{F}_{\overline{u}} = \left\lbrace \operatorname{\mathrm{rank}}\left(\begin{array}{ccc} \overline{u}_1 & \overline{u}_2 & \overline{u}_3 \\ B_1x_0^2+B_2x_0x_1+B_3x_1^2 & B_2x_0^2+B_3x_0x_1 & B_3x_0^2 \end{array}\right) < 2\right\rbrace \subset \mathbb{P}^1_{(x_0,x_1)}\times \mathbb{P}^{n}_{(w_1,\dots, w_{n+1})}. \end{align*} $$

So $\widetilde {F}_{\overline {u}}$ is unirational and maps dominantly onto $\mathbb {P}^1_{(x_0,x_1)}$ . Finally, to conclude it is enough to note that $Y_{(3,2)}\rightarrow \mathbb {P}^1_{(x_0,x_1)}$ is a fibration in quadric hypersurfaces and to apply Proposition 2.7 and Remark 2.8.

Proposition 3.13. Let $X_4\subset \mathbb {P}^{n+1}$ be a quartic hypersurface having double points along a codimension two linear subspace $\Lambda \subset \mathbb {P}^{n+1}$ , with a point $p\in X_4\setminus \Lambda $ and otherwise general. If $n\geq 4$ , then $X_4$ is unirational.

Proof. The equation of $X_4\subset \mathbb {P}^{n+1}$ can be written as in Lemma 3.9 for $d = 4$ . Hence, the claim follows from Lemma 3.9 and Proposition 3.10.

For quartic $3$ -folds, we have the following density result.

Proposition 3.14. Let $X_4\subset \mathbb {P}^{4}$ be a quartic hypersurface, over a number field k, having double points along a codimension two linear subspace $\Lambda \subset \mathbb {P}^{4}$ , with a point $p\in X_4\setminus \Lambda $ and otherwise general. The set $X_4(k)$ of the k-rational points of $X_4$ is Zariski dense in $X_4$ .

Proof. By Lemma 3.9, it is enough to prove that a general divisor $Y_{3,2}\subset \mathbb {P}^1\times \mathbb {P}^3$ as in Proposition 3.10 has dense k-points. Consider the $2$ -plane $H = \{x_1 = w_1 = 0\}\subset Y_{3,2}$ and the rational map

induced by (3.11). Note that $\eta '$ maps H dominantly onto $\mathbb {P}^{2}_{(u_1,u_2,u_{3})}$ . Take a general point $p\in H$ , its image $q = \eta '(p)$ and a general line $L = \{\alpha _1u_1+\alpha _2u_2+\alpha _3u_3 = 0\}\subset \mathbb {P}^{2}_{(u_1,\dots ,u_{3})}$ through q. Set $S_{L,p} = \eta ^{\prime -1}(L)$ . Since $Y_{(3,2)}$ is general $\eta '$ restricts to a morphism $\eta ^{\prime }_{|S_{L,p}}:S_{L,p}\rightarrow \mathbb {P}^1$ . A straightforward computation shows that $S_{L,p}$ is smooth, and by the argument in the second part of the proof of Proposition 3.10, the general fiber of $\eta ^{\prime }_{|S_{L,p}}:S_{L,p}\rightarrow \mathbb {P}^1$ is a smooth curve of genus one.

Furthermore, $C = H\cap S_{L,p} = \{\alpha _1B_1+\alpha _2B_2+\alpha _3B_3 = x_1 = w_1 = 0\}$ . Hence, since $Y_{3,2}$ is general, C is a smooth conic with a point $p\in C$ . Note that a general fiber of $\eta ^{\prime }_{|S_{L,p}}:S_{L,p}\rightarrow \mathbb {P}^1$ intersects a general fiber of the quadric bundle $Y_{3,2}\rightarrow \mathbb {P}^1$ in a zero-dimensional scheme of degree eight. So, $C\subset S_{L,p}$ is a rational $4$ -section of $\eta ^{\prime }_{|S_{L,p}}$ .

Consider the fiber product $T = C\times _{L}S_{L,p}$ . Since $p,L$ and $Y_{3,2}$ are general, the ramification divisor of $\eta ^{\prime }_{|C}:C\rightarrow \mathbb {P}^1$ is disjoint from the singular fibers of $\eta ^{\prime }_{|S_{L,p}}$ . Hence, T is smooth and the proof of [HT00, Theorem 8.1] goes through. So the set $S_{L,p}(k)$ of the k-rational points of $S_{L,p}$ is Zariski dense in $S_{L,p}$ . Finally, letting the point p vary in H and the line L vary among the lines passing through $q = \eta '(p)$ we get the claim.

The following is our main result on the unirationality of quadric hypersurfaces having double points along a linear subspace.

Theorem 3.15. Let $X_4\subset \mathbb {P}^{n+1}$ be a quartic hypersurface having double points along an h-plane $\Lambda \subset \mathbb {P}^{n+1}$ , with a point $p\in X_4\setminus \Lambda $ and otherwise general. If $n\geq 4$ and $h\geq 3$ , then $X_4$ is unirational.

Furthermore, if k is a $C_r$ field, $n\geq 3$ , and $s+1> 2^r$ , where $s = \max \{n-h,h\}$ a general quartic hypersurface $X_4\subset \mathbb {P}^{n+1}$ having double points along an h-plane is unirational.

Proof. The case $h = n-1$ comes from Proposition 3.13. Let be the projection from $H = \left \langle p,\Lambda \right \rangle $ , $\widetilde {\pi }_{H}:\widetilde {X}_4\rightarrow \mathbb {P}^{n-h-1}$ , and $F_p\cong \mathbb {P}^{n-h-1}$ the fiber over p of the blowup $\widetilde {X}_4\rightarrow X_4$ along $H\cap X_4$ . Note that since $X_4$ is irreducible the blowup $\widetilde {X}_4$ is also irreducible.

The generic fiber $\widetilde {X}_{4,\eta }$ of $\widetilde {\pi }_{H}:\widetilde {X}_4\rightarrow \mathbb {P}^{n-h-1}$ is a quartic hypersurface $\widetilde {X}_{4,\eta }\subset \mathbb {P}^{h+2}_{k(t_1,\dots ,t_{n-h-1})}$ with double points along an h-plane and with a $k(t_1,\dots ,t_{n-h-1})$ -rational point induced by $F_p$ .

Therefore, Proposition 3.13 yields that $\widetilde {X}_{4,\eta }$ is unirational over $k(t_1,\dots ,t_{n-h-1})$ and since $\widetilde {X}_4$ is irreducible $\widetilde {X}_4$ is unirational.

Now, assume k to be $C_r$ . The exceptional divisor $\widetilde {E}$ in Lemma 2.4 is a divisor of bidegree $(2,2)$ in $\mathbb {P}^{n-h}\times \mathbb {P}^h$ , and since $X_4$ is general $\widetilde {E}$ maps dominantly onto $\mathbb {P}^{n-h}$ . Furthermore, since both the fibrations on $\widetilde {E}$ have quadric hypersurfaces as fibers and $s+1> 2^r$ by Remark 2.10 the divisor $\widetilde {E}$ has a point. To conclude, it is enough to apply [Mas22, Theorem 1.8] and Proposition 2.7.

Remark 3.16. In the second part of Theorem 3.15, the assumption on the base field is needed in order to ensure the existence of a point in the exceptional divisor $\widetilde {E}\subset \mathbb {P}^{n-h}\times \mathbb {P}^h$ . Indeed, over non $C_r$ fields there are divisors of bidegree $(2,2)$ in $\mathbb {P}^{n-h}\times \mathbb {P}^h$ without points [Mas22, Remark 4.15].

Remark 3.17. In Theorem 3.15, the assumption on the existence of a point in $X_4\setminus \Lambda $ is necessary as the following argument shows. Take, for instance, $n = 3$ , $h = 1$ and $k = \mathbb {R}$ . By Lemma 2.4, $X_4$ is birational to a conic bundle $D\subset \mathcal {T}_{1,0,0}$ of multidegree $(4,3,3,2,2,2)$ . We may write D as the zero set of a polynomial

$$ \begin{align*}f = a_0x_0^4 + a_1x_0^3x_1+a_2x_0^2x_1^2+a_3x_0x_1^3+a_4x_1^4, \end{align*} $$

where $a_i\in \mathbb {R}[x_1,x_2,y_0,y_1,y_2]$ . Let V be the $\mathbb {R}$ -vector space parametrizing these conic bundles and consider the map

$$ \begin{align*}\begin{array}{cccc} ev: & V\times\mathbb{R}^5 & \longrightarrow & \mathbb{R}^5 \\ & (f,(\overline{x}_1,\overline{x}_2,\overline{y}_0,\overline{y}_1,\overline{y}_2)) & \mapsto & P_{f,\overline{x}_1,\overline{x}_2,\overline{y}_0,\overline{y}_1,\overline{y}_2}(x_0), \end{array} \end{align*} $$

where $P_{f,\overline {x}_1,\overline {x}_2,\overline {y}_0,\overline {y}_1,\overline {y}_2}(x_0) = f(x_0,\overline {x}_1,\overline {x}_2,\overline {y}_0,\overline {y}_1,\overline {y}_2)\in \mathbb {R}[x_0]$ . The polynomials $P_{f,\overline {x}_1,\overline {x}_2,\overline {y}_0,\overline {y}_1,\overline {y}_2}(x_0)$ with a real root form a semialgebraic subset $R_{rr}\subset \mathbb {R}^5$ of maximal dimension; see, for instance, [BPR06, Section 4.2.3]. Hence, $ev^{-1}(R_{rr})\subset V\times \mathbb {R}^5$ is also a semialgebraic subset of maximal dimension and then the Tarski–Seidenberg principle [BCR98, Section 2.2] yields that $Z_{rp}= \pi _1(ev^{-1}(R_{rr}))\subset V$ , where $\pi _1:V\times \mathbb {R}^5\rightarrow V$ is the projection, is a semialgebraic subset of maximal dimension. Note that

$$ \begin{align*}Z_{rp} = \{D\subset\mathcal{T}_{1,0,0} \text{ of mutidegree } (4,3,3,2,2,2) \text{ having a rational point}\}. \end{align*} $$

The complementary set $Z_{nrp} = Z_{rp}^c$ is nonempty, take, for instance,

$$ \begin{align*}D = \{(x_0^2+x_1^2+x_2^2)^2y_0^2+(x_0^2+x_1^2+2x_2^2)y_1^2+(x_0^2+x_1^2+3x_2^2)y_2^2 = 0\}\subset \mathcal{T}_{1,0,0}. \end{align*} $$

Hence, $Z_{nrp}\subset V$ is also a semialgebraic subset of maximal dimension. By Lemma 2.4, the quartics $X_4\subset \mathbb {P}^4$ corresponding to the conic bundles in $Z_{nrp}$ do not have a point in $X_4\setminus \Lambda $ , and hence, they cannot be unirational.

Lemma 3.18. Consider the hypersurface

$$ \begin{align*}X_d = \{z_0^{d-2}Q + z_0^{d-3}z_1 A_1+\dots + z_0z_1^{d-3}A_{d-3}+z_1^{d-2}A_{d-2} = 0\}\subset\mathbb{P}^{n+1}_{(z_0,\dots,z_{n+1}),} \end{align*} $$

where $A_i = A_i^2 + z_0A_i^1+z_0^2A_i^0$ , and $A_i^1\in k[z_1,\dots ,z_{n+1}]_1$ , $Q,A_i^2\in k[z_1,\dots ,z_{n+1}]_2$ , $A_i^0\in k$ for $i = 1,\dots ,d-2$ . Then $X_d$ is birational to the divisor

$$ \begin{align*}Y_{(d-2,2)} = \left\{ x_0^{d-2}Q+\sum_{i=1}^{d-2}x_0^{d-2-i}x_1^iA_i = 0\right\}\subset\mathbb{P}^1_{(x_0,x_1)}\times \mathbb{P}^{n}_{(z_1,\dots, z_{n+1})} \end{align*} $$

of bidegree $(d-2,2)$ .

Proof. It is enough to substitute $z_0 = \frac {x_0}{x_1}z_1$ in the equation of $X_d$ and to clear the denominators of the resulting polynomial.

For quartic hypersurfaces with a double point, we get a version of Theorem 3.15 with a slightly less restrictive condition on n.

Proposition 3.19. Let $X_d\subset \mathbb {P}^{n+1}$ be a hypersurface of degree d having multiplicity $d-2$ along a codimension two linear subspace $\Lambda \subset \mathbb {P}^{n+1}$ , with a point $p\in X_d\setminus \Lambda $ , a double point $q\in X_d\setminus \Lambda $ and otherwise general. If either $d = 4$ and $n\geq 2$ or $d = 5$ and $n\geq 4$ , then $X_d$ is unirational.

Proof. We may assume that $\Lambda = \{z_0 = z_1 = 0\}$ and $q = [1:0:\dots :0]$ so that the equation of $X_d\subset \mathbb {P}^{n+1}$ can be written as in Lemma 3.18. First, consider the case $d = 4$ . By Lemma 3.18 $X_4$ is birational to a divisor $Y_{(2,2)}\subset \mathbb {P}^1_{(x_0,x_1)}\times \mathbb {P}^{n}_{(z_1,\dots , z_{n+1})}$ of bidegree $(2,2)$ . The point $p\in X_d\setminus \Lambda $ yields a point $p'\in Y_{(2,2)}$ . Let H be a general $2$ -plane in $\mathbb {P}^{n}_{(z_1,\dots , z_{n+1})}$ through the projection of $p'$ and set $\Gamma = \mathbb {P}^1_{(x_0,x_1)}\times H$ . Then $W = Y_{(2,2)}\cap \Gamma $ is a divisor of bidegree $(2,2)$ in $\mathbb {P}^1_{(x_0,x_1)} \times \mathbb {P}^2$ which by [KM17, Corollary 8] is unirational. Since W maps dominantly onto $\mathbb {P}^1_{(x_0,x_1)}$ , to conclude it is enough to apply Proposition 2.7.

Now, let $d = 5$ . Then by Lemma 3.18 $X_5$ is birational to a divisor $Y_{(3,2)}\subset \mathbb {P}^1_{(x_0,x_1)}\times \mathbb {P}^{n}_{(z_1,\dots , z_{n+1})}$ of bidegree $(3,2)$ . This divisor is not of the same form of those considered in Proposition 3.10. However, the argument in the proof of Proposition 3.10 goes through smoothly and shows $Y_{(3,2)}$ is unirational.

Remark 3.20. The argument in the proof of Proposition 3.10 works indeed for a general divisor of bidegree $(3,2)$ as shown in [Mas22, Proposition 3.2] which works for a divisor of bidegree $(3,2)$ in $\mathbb {P}^1\times \mathbb {P}^n$ with a point and otherwise general. So [Mas22, Proposition 3.2] cannot be applied directly to the divisors $Y_{(3,2)}$ in Proposition 3.10 which are special. In the proof of Proposition 3.10, it is shown that the argument in the proof of [Mas22, Proposition 3.2] works for a general $Y_{(3,2)}$ , in other words that a general divisor of the form $Y_{(3,2)}$ is general enough, among the divisor of bidegree $(3,2)$ in $\mathbb {P}^1\times \mathbb {P}^n$ , for [Mas22, Proposition 3.2] to apply.

Lemma 3.21. Let $D\subset \mathbb {P}^{n-h}\times \mathbb {P}^h$ be an irreducible divisor of bidegree $(1,a)$ . Then D is rational.

Proof. If $n-h = 1$ , then the projection $\pi _{2|D}:D\rightarrow \mathbb {P}^h$ is birational. Assume $n-h\geq 2$ . The generic fiber of $\pi _{2|D}:D\rightarrow \mathbb {P}^h$ is an $(n-h-1)$ -dimensional projective space over the function field of $\mathbb {P}^h$ . So we can find an open subset $U\subset \mathbb {P}^h$ such that $\pi _{2|D}^{-1}(U)$ is birational to $U\times \mathbb {P}^{n-h-1}$ . Finally, since D is irreducible we conclude that D is rational.

The following is our main result on the unirationality of quartic hypersurface containing a linear subspace.

Theorem 3.22. Let $X_4\subset \mathbb {P}^{n+1}$ be an irreducible quartic hypersurface containing an h-plane $\Lambda \subset \mathbb {P}^{n+1}$ . Assume that $\Lambda \cap \operatorname {\mathrm {Sing}}(X_4)$ has dimension at most $h-2$ and that $X_4$ is not a cone over a quartic of smaller dimension. If $n\geq 3$ and $h\geq 2$ , then $X_4$ is unirational.

Proof. Consider the blowup $\widetilde {X}_4$ along $\Lambda $ in Lemma 2.4. Under our hypotheses, $X_4$ cannot be singular at the general point of $\Lambda $ . So the exceptional divisor $\widetilde {E}\subset \mathbb {P}^{n-h}\times \mathbb {P}^h$ has bidegree $(1,3)$ , and Lemma 3.21 yields that $\widetilde {E}$ is unirational. The projection $\pi :\widetilde {X}_4\rightarrow \mathbb {P}^{n-h}$ is a fibration in h-dimensional cubic hypersurfaces. Therefore, thanks to Proposition 2.7 and Remark 2.8 to conclude it is enough to show that $\pi _{|\widetilde {E}}:\widetilde {E}\rightarrow \mathbb {P}^{n-h}$ is dominant.

Since $X_4$ is not a cone over a quartic of smaller dimension by the expression of $\widetilde {X}_4$ in (2.6), we see that $\pi _{|\widetilde {E}}:\widetilde {E}\rightarrow \mathbb {P}^{n-h}$ is not dominant if and only if the equation of $X_d$ is of the following form

$$ \begin{align*}X_d = \left\{\sum_{i=0}^{n-h}z_i(C_i(z_0,\dots,z_{n-h})+c_iP) = 0\right\}\subset\mathbb{P}^{n+1}, \end{align*} $$

where $C_i\in k[z_0,\dots ,z_{n-h}]_{d-1}$ , $c_i\in k$ and $P\in k[z_0,\dots ,z_{n+1}]_{d-1}$ . In particular,

$$ \begin{align*}Z = \{z_0 = \dots = z_{n-h} = P = 0\}\subset \Lambda\cap\operatorname{\mathrm{Sing}}(X_d)\end{align*} $$

and since $\dim (Z)\geq h-1$ we get a contradiction.

Corollary 3.23. Assume that $h\geq 2$ , $n> 2h-1$ , and let $X_4\subset \mathbb {P}^{n+1}$ be a general quartic hypersurface containing an h-plane. Then $X_4$ is unirational.

Proof. Note that $n> 2h-1$ yields that a general quartic containing a fixed h-plane is smooth. The claim follows from Theorem 3.22.

4 Divisors in products of projective spaces and Quintics

We now study the unirationality of quintic hypersurfaces that are singular along linear subspaces and of divisors of bidegree $(3,2)$ in products of projective spaces. We begin with a series of preliminary results that we will need later on.

Lemma 4.1. Let $\mathcal {C}\rightarrow W$ be a fibration in m-dimensional cubic hypersurfaces with $m\geq 2$ , $\mathcal {C}$ irreducible and W a rational variety over a $C_r$ field k. If the general fiber of $\mathcal {C}\rightarrow W$ is an irreducible cubic with no triple point and

$$ \begin{align*}m> 3^{r+\dim(W)} - 2, \end{align*} $$

then $\mathcal {C}$ is unirational.

Proof. Let $\mathcal {C}'$ be the generic fiber of $\mathcal {C}\rightarrow W$ . Then $\mathcal {C}'$ is a cubic hypersurface over the function field $F = k(t_1,\dots ,t_{\dim (W)})$ . Since $m+2> 3^{r+\dim (W)}$ , Remark 2.10 yields that $\mathcal {C}'$ has a point. Hence, by [Kol02, Theorem 1] $\mathcal {C}'$ is unirational over F and so since $\mathcal {C}$ is irreducible $\mathcal {C}$ is unirational.

Lemma 4.2. Let $Q^{N-1}\subset \mathbb {P}^{N}$ be a smooth $(N-1)$ -dimensional quadric hypersurface over a $C_r$ field. If

$$ \begin{align*}N-2s+1> 2^r \end{align*} $$

for some $0\leq s \leq \lfloor \frac {N-1}{2}\rfloor $ , then through any point of $Q^{N-1}$ there is an s-plane contained in $Q^{N-1}$ .

Proof. Since $N-1> 2^r - 2$ by Remark 2.10 $Q^{N-1}$ has a point. Hence, $Q^{N-1}$ is rational and in particular the set of its rational points is dense. Take a point $x_0 \in Q^{N-1}$ .

The tangent space $T_{x_0}Q^{N-1}$ cuts out on $Q^{N-1}$ a cone with vertex $x_0$ over an $(N-3)$ -dimensional quadric $Q^{N-3}$ . Since $N-3> 2^r - 2$ Remark 2.10 yields the existence of a point $x_1 \in Q^{N-3}$ . The line $\left \langle x_0,x_1\right \rangle $ is therefore contained in $Q^{N-1}$ .

Now, $T_{x_1}Q^{N-3}\cap Q^{N-3}$ is a cone with vertex $x_1$ over an $(N-5)$ -dimensional quadric $Q^{N-5}$ which again by Remark 2.10 has a point $x_2\in Q^{N-5}$ . The $2$ -plane $\left \langle x_0,x_1,x_2\right \rangle $ is contained in $Q^{N-1}$ .

Proceeding recursively, we have that $T_{x_{s-1}}Q^{N-2(s-1)-1}\cap Q^{N-2(s-1)-1}$ is a cone with vertex $x_{s-1}$ over an $(N-2s-1)$ -dimensional quadric $Q^{N-2s-1}$ . Since $N-2s+1> 2^r$ , the quadric $Q^{N-2s-1}$ has a point $x_s$ and the s-plane $\left \langle x_0,\dots ,x_s\right \rangle $ is then contained in $Q^{N-1}$ .

Lemma 4.3. Let $X = Q_1\cap \dots \cap Q_c\subset \mathbb {P}^N$ be a smooth complete intersection of quadrics. Assume that X contains a $(c-1)$ -plane $\Lambda $ . Then X is rational.

Proof. Note that since X is smooth $Q_i$ must be smooth along X. Let H be a general c-plane containing $\Lambda $ . Then H intersects $Q_i$ along $\Lambda \cup H_i$ where $H_i$ is a $(c-1)$ -plane. Hence, we get c linear subspaces of dimension $c-1$ of $H\cong \mathbb {P}^c$ .

Since H is general, these $(c-1)$ -planes intersect in a point $x_H = H_1\cap \dots \cap H_c \in X$ . To conclude, it is enough to parametrize X with the $\mathbb {P}^{N-c}$ of c-planes containing $\Lambda $ .

Proposition 4.4. Let $X = Q_1\cap \dots \cap Q_c\subset \mathbb {P}^{N}$ be a smooth complete intersection of quadrics over a $C_r$ field. If

$$ \begin{align*}N-s(c+1)+1> 2^rc, \end{align*} $$

then through any point of X there is an s-plane contained in X.

Proof. For $s = 0$ , the claim follows from Remark 2.10. We proceed by induction on s. Since

$$ \begin{align*}N-(s-1)(c+1)+1> N-s(c+1)+1 > 2^rc \end{align*} $$

through any point of X, there is an $(s-1)$ -plane. Fix one of these $(s-1)$ -planes, and denote it by $\Lambda ^{s-1}\subset X$ . The s-planes in $\mathbb {P}^{N}$ containing $\Lambda ^{s-1}$ are parametrized by $\mathbb {P}^{N-s}$ .

Arguing as in the proof of Lemma 4.2, we see that requiring such an s-plane to be contained in $Q_i$ yields s linear equations given by the tangent spaces of $Q_i$ at s general points of $\Lambda ^{s-1}$ plus a quadratic equation induced by $Q_i$ itself.

Hence, the points of $\mathbb {P}^{N-s}$ corresponding to s-planes contained in X are parametrized by a subvariety cut out by $sc$ linear equations and c quadratic equations that is an intersection Y of c quadrics in $\mathbb {P}^{N-s-sc}$ . Since $\dim (Y) = N-s(c+1)-c> 2^rc-1-c \geq -1$ and $N-s(c+1)+1> 2^rc$ , Remark 2.10 yields that Y has a point and so there is an s-plane through $\Lambda ^{s-1}$ contained in X.

Corollary 4.5. Let $X = Q_1\cap \dots \cap Q_c\subset \mathbb {P}^{N}$ be a smooth complete intersection of quadrics over a $C_r$ field. If

$$ \begin{align*}N-c^2+2> 2^rc, \end{align*} $$

then X is rational.

Proof. By Proposition 4.4 with $s = c-1$ , the complete intersection X contains a $(c-1)$ -plane, and hence, the claim follows from Lemma 4.3.

We are now ready to prove our first result on unirationality of quintic hypersurfaces.

Proposition 4.6. Let $X^{n}_5 = \{\overline {f} = 0\}\subset \mathbb {P}^{n+1}$ be a quintic hypersurface of the form

$$ \begin{align*}\overline{f} = \sum_{ i_0+\dots +i_{n-h} = 3}x_0^{i_0}\dots x_{n-h}^{i_{n-h}}A_{i_0,\dots,i_{n-h}}(y_0,\dots,y_h,x_{n-h}) \end{align*} $$

with $A_{i_0,\dots ,i_{n-h}}$ quadratic polynomials. If either

1. (i) $h\geq 4$ and the complete intersection

   $$ \begin{align*}W' = \{A_{3,\dots,0} = A_{2,1,0,\dots,0} = x_0 = \dots = x_{n-h} = 0\}\subset\mathbb{P}^{h}_{(y_0,\dots,y_h)} \end{align*} $$

   contains a line, or

2. (ii) $h\geq 5$ and the quadric

   $$ \begin{align*}\widetilde{Q} = \{x_0 = \dots = x_{n-h} = A_{3,\dots,0} = 0\}\subset\mathbb{P}^{h}_{(y_0,\dots,y_h)} \end{align*} $$

   contains a $2$ -plane,

and $X^{n}_5$ is otherwise general, then $X_5^{n}$ is unirational.

Proof. Note that $X^{n}_5$ is a hypersurface of the form (2.12) in Proposition 2.11. Hence, $X^{n}_5$ has multiplicity three along $\Lambda = \{x_0 = \dots = x_{n-h} = 0\}$ , and multiplicity two along $\Lambda ' = \{y_0 = \dots = y_h = x_{n-h} = 0\}$ .

First, consider $\text {(i)}$ . Fix a point $p \in \Lambda '$ , say $p = [1:0:\dots : 0]$ . The lines through p intersecting $X_5^n$ at p with multiplicity at least four are parametrized by the variety

$$ \begin{align*}Y = \{A_{3,\dots,0} = x_1A_{2,1,0,\dots,0} + \dots + x_{n-h}A_{2,0,\dots,0,1} = 0\}\subset\mathbb{P}^{n}_{(x_1,\dots,x_{n-h},y_0,\dots,y_h)}.\end{align*} $$

Assume Y to be unirational. Associating to a general point $y\in Y$ , representing a line $l_y$ , the fifth intersection point of $l_y$ and $X_5^n$ we get a rational map

Set $Y_{\psi } = \overline {\psi (Y)}\subset X_5^n$ , and let

be the restriction to $X_5^n$ of the projection from $\Lambda $ . We want to prove that

is dominant. Indeed, if so $Y_{\psi }$ would be a unirational variety transverse to the quadric fibration induced by the projection from $\Lambda $ and dominating $\mathbb {P}^{n-h}$ , and Proposition 2.7 would imply that $X_5^n$ is unirational.

Since $Y_{\psi }$ is unirational it is enough to prove that the induced map between the algebraic closures is dominant. Consider the complete intersection

$$ \begin{align*}Z = Y\cap X^n_5 = \{A_{3,\dots,0} = x_1A_{2,1,0,\dots,0} + \dots + x_{n-h}A_{2,0,\dots,0,1} = \overline{f}= 0\}\subset \mathbb{P}^{n+1}_{(x_0,\dots,x_{n-h},y_0,\dots,y_h)}. \end{align*} $$

A general point of $z\in Z\subset X^n_5$ represents a line $l_z$ intersecting $X^n_5$ with multiplicity four at p and multiplicity one at z. Hence, $Z\subset Y_{\psi }$ .

Fix a general point $q\in \mathbb {P}^{n-h}$ , and consider

$$ \begin{align*}Z_q = \overline{\pi}_{|Y_{\psi}}^{-1}(q)\cap Z.\end{align*} $$

Note that since $q\in \mathbb {P}^{n-h}$ is defined by $n-h$ equations, $Z\subset \mathbb {P}^{n+1}_{(x_0,\dots ,x_{n-h},y_0,\dots ,y_h)}$ is cut out by three equations and by hypothesis $n+1-(n-h)-3 = h-2> 0$ . So $Z_q$ is nonempty. Then $\overline {\pi }_{|W_{\psi }}$ is dominant when restricted to $\overline {Z}$ , and hence, it is dominant.

Now, we prove that Y is unirational. In $\mathbb {P}^{n}_{(x_1,\dots ,x_{n-h},y_0,\dots ,y_h)}$ , fix the point $p' = [1:0:\dots :0]$ . The quadric

$$ \begin{align*}\{x_1 = \dots = x_{n-h} = A_{2,1,0,\dots,0}\}\subset\mathbb{P}^{h}_{(y_0,\dots,y_h)}\end{align*} $$

parametrizes lines that are contained in

$$ \begin{align*}Y_3 = \{x_1A_{2,1,0,\dots,0} + \dots + x_{n-h}A_{2,0,\dots,0,1}\}.\end{align*} $$

Indeed, the tangent cone of $Y_3$ at $p'$ is defined by $\{A_{2,1,0,\dots ,0} = 0\}$ and $\{x_1 = \dots = x_{n-h}= 0\}\subset Y_3$ . So these lines intersect $Y_3$ with multiplicity three at $p'$ and at another point in the linear space $\{x_1 = \dots = x_{n-h}= 0\}$ .

Set $Y_2 = \{A_{3,\dots ,0} = 0\}$ , and let W be the cone over $W'$ with vertex $p'$ . Since $p'$ is in the vertex of $Y_2$ , we have that

$$ \begin{align*}W\subset Y = Y_2\cap Y_3.\end{align*} $$

Furthermore, since $W'$ contains a line by Lemma 4.3 it is rational, and hence, W is rational as well.

As in the proof of Proposition 3.4, we construct a quadric bundle $\widetilde {\mathcal {Q}}\rightarrow W$ with $(n-4)$ -dimensional fibers whose general point $(w,l_w)$ represents a point $w\in W$ and a line $l_w$ which is contained in $Y_2$ and intersects $Y_3$ with multiplicity two at w.

Associating to $(w,l_w)$ the third point of intersection of $l_w$ and $Y_3$ , we get a rational map and arguing as in the proof of Proposition 3.4 we see that such rational map is dominant.

Now, we consider $\text {(ii)}$ . Note that

$$ \begin{align*}\widetilde{Q}' = \{x_1 = \dots = x_{n-h} = A_{3,\dots,0} = 0\}\subset X^n_5\subset\mathbb{P}^{n+1} \end{align*} $$

is an h-dimensional quadric cone over $\widetilde {Q}$ with vertex $[1:0:\dots :0]$ . Since $\widetilde {Q}$ contains a $2$ -plane $\widetilde {Q}'$ contains a $3$ -plane $H\subset X^n_5$ . If $x\in H$ is a general point the lines through x intersecting $X^n_5$ with multiplicity four at x are parametrized by a complete intersection of a quadric and a cubic in $\mathbb {P}^{n-1}$ . Hence, we get a fibration $\mathcal {Y}_{2,3}\rightarrow H$ whose fiber over a general $x\in H$ is a complete intersection $Y_{2,3,x}$ of a quadric $Y_{2,x}\subset \mathbb {P}^{n-1}$ and a cubic $Y_{3,x}\subset \mathbb {P}^{n-1}$ . Note that since $X_5^n$ is general among the quintics satisfying $(ii)$ for $x\in H$ both $Y_{2,x}$ and $Y_{2,3,x}$ are smooth.

The generic fiber of $\mathcal {Y}_{2,3}\rightarrow H$ is then a complete intersection $\mathcal {Y}_{2,3,k(H)} = \mathcal {Y}_{2,k(H)}\cap \mathcal {Y}_{3,k(H)}$ satisfying the hypotheses of Lemma 3.1. Indeed, by considering the $2$ -plane parametrizing lines in H through a general $x\in H$ we get a $2$ -plane over $k(H)$ contained in $\mathcal {Y}_{2,k(H)}$ .

Therefore, by Lemma 3.1 $\mathcal {Y}_{2,3,k(H)}$ is unirational over $k(H)$ . As usual after replacing H by an open subset, we may assume that $\mathcal {Y}_{2,3}$ is irreducible and so we get that $\mathcal {Y}_{2,3}$ is unirational. Now, a general point of $\mathcal {Y}_{2,3}$ represents a pair $(x,l_x)$ with $x\in H$ general and $l_x$ a line intersecting $X_5^n$ with multiplicity four at x. As usual, associating to $(x,l_x)\in \mathcal {Y}_{2,3}$ the fifth point of intersection of $l_x$ with $X_5^n$ we get a rational map

Note that $\dim (\mathcal {Y}_{2,3}) = 3+(n-3) = n$ . Arguing as in the last part of the proof of Proposition 3.4, we see that $\phi $ is generically finite and therefore it is dominant.

Theorem 4.7. Let $D\subset \mathbb {P}^{n-h}\times \mathbb {P}^{h+1}$ , with $n-h\geq 2$ , be a general divisor of bidegree $(3,2)$ over a $C_r$ field. If either

1. (i) $n\geq 4$ , $2h\geq n+3$ and $h> 2^{r+1}+2$ , or

2. (ii) $n\geq 6$ , $h\geq 5$ and $h> 2^{r}+3$ , or

3. (iii) $n-h-1> 3^{r+1}-2$ ,

then D is unirational.

Proof. By Proposition 2.11, D is birational to a quintic hypersurface $X^n_{5}\subset \mathbb {P}^{n+1}$ of the form considered in Proposition 4.6.

Consider $\text {(i)}$ . Since, $h> 2^{r+1}+2$ by Proposition 4.4 the complete intersection $W'$ in Proposition 4.6 contains a line, and hence, Proposition 4.6 yields that $X^n_{5}$ is unirational.

For $\text {(ii)}$ , note that since $h> 2^{r}+3$ Lemma 4.2 implies that the quadric $\widetilde {Q}$ in Proposition 4.6 contains a $2$ -plane.

Now, consider $\text {(iii)}$ . The projection $\pi _2:D\rightarrow \mathbb {P}^{h+1}$ endows D with a structure of fibration in $(n-h-1)$ -dimensional cubic hypersurfaces. Take a general line $L\subset \mathbb {P}^{h+1}$ , and set $\mathcal {C}_L = \pi _2^{-1}(L)$ . Then $\mathcal {C}_L$ is a fibration in $(n-h-1)$ -dimensional cubic hypersurfaces over $\mathbb {P}^1$ and since $n-h-1> 3^{r+1}-2$ Lemma 4.1 yields that $\mathcal {C}_L$ is unirational. Finally, to conclude it is enough to note that $\pi _{|\mathcal {C}_L}:\mathcal {C}_L\rightarrow \mathbb {P}^{n-h}$ is dominant and to apply Proposition 2.7.

Remark 4.8. Take for instance $r = 0$ that is k is algebraically closed. Remark 2.10 gives the rationality of a $D\subset \mathbb {P}^{n-h}\times \mathbb {P}^{h+1}$ as in Theorem 4.7 for $h> 2^{n-h}-2$ while Theorem 4.7 gives the unirationality of D for $h> 4$ as long as $n\geq 4$ and $h \geq n-h+3$ . For example, take $h = 10$ . Then Remark 2.10 yields that D is rational for $n\leq 13$ while Theorem 4.7 gives the unirationality of D for $n\leq 17$ .

Furthermore, there are cases covered by $\text {(iii)}$ but not by $\text {(i)}$ . For instance, by $\text {(iii)}$ we get that a general $D\subset \mathbb {P}^{n-1}\times \mathbb {P}^{2}$ of bidegree $(3,2)$ and dimension at least four is unirational.

For $r\geq 1$ , $\text {(i)}$ generally performs better that $\text {(ii)}$ . For instance, the case $h = 7, n = 11$ is covered by $\text {(i)}$ but not by $\text {(ii)}$ .

Remark 4.9. In particular, Theorem 4.7 (ii) together with [ABP18, Theorem A] yields that a very general, meaning outside of a countable union of closed subsets of the corresponding parameter space, divisor of bidegree $(3,2)$ in $\mathbb {P}^3\times \mathbb {P}^2$ , over an algebraically closed field of characteristic zero, is unirational but not stably rational.

We end this section with our main results on the unirationality of quintic hypersurfaces which are singular along a linear subspace.

Theorem 4.10. Let $X_5\subset \mathbb {P}^{n+1}$ be a quintic hypersurface over a $C_r$ field having multiplicity three along an h-plane and otherwise general. Assume that $n-h \geq 2$ . If either

1. (i) $n\geq 5$ , $2h\geq n+4$ and $h> 2^{r+1}+3$ , or

2. (ii) $n\geq 7$ , $h\geq 6$ and $h> 2^{r}+4$ , or

3. (iii) $n-h-1> 3^{r+1}-2$ ,

then $X_5$ is unirational.

Similarly, if $X_5\subset \mathbb {P}^{n+1}$ has multiplicity two along an h-plane with $h\geq 2$ and is otherwise general, and either

1. (i) $n\geq 5$ , $2h\leq n-4$ and $n-h> 2^{r+1}+3$ , or

2. (ii) $n\geq 7$ , $n-h-1\geq 5$ and $n-h-1> 2^{r}+3$ , or

3. (iii) $h> 3^{r+1}-1$ ,

then $X_5$ is unirational.

Proof. By Lemma 2.4, the exceptional divisor $\widetilde {E}\subset \widetilde {X}_5$ is a divisor of bidegree $(3,2)$ in $\mathbb {P}^{n-h}\times \mathbb {P}^h$ . Since $X_5$ is general, $\widetilde {E}$ maps dominantly onto $\mathbb {P}^{n-h}$ , and hence, to conclude we apply Proposition 2.7 and Theorem 4.7. For the second statement, it is enough to argue as in the previous case on a divisor of bidegree $(2,3)$ in $\mathbb {P}^{n-h}\times \mathbb {P}^h$ .

Proposition 4.11. Let $X_5\subset \mathbb {P}^{n+1}$ be a quintic hypersurface over a field k having multiplicity three along an $(n-1)$ -plane and otherwise general. If either

1. (i) $n\geq 5$ and k is either a number field or a real closed field, or

2. (ii) k is $C_r$ , $n> 4$ and $n> 2^r$ ,

then $X_5$ is unirational.

Proof. The exceptional divisor $\widetilde {E}\subset \widetilde {X}_5$ is a general divisor of bidegree $(3,2)$ in $\mathbb {P}^{1}\times \mathbb {P}^{n-1}$ . By [Mas22, Corollary 4.13, Lemma 4.18] and Remark 2.10, under our hypotheses $\widetilde {E}$ has a point and hence [Mas22, Theorem 1.8] yields that $\widetilde {E}$ is unirational. To conclude, it is enough to argue as in the proof of Theorem 4.10 applying Propositions 2.7.

Remark 4.12. When k is a real closed field, the unirationality of a quintic hypersurface $X_5\subset \mathbb {P}^{n+1}$ as in Proposition 4.11 follows from [Kol99, Corollary 1.8].

Proposition 4.13. Let $X_d\subset \mathbb {P}^{n+1}$ be a hypersurface of degree d over a field k having multiplicity $d-2$ along an h-plane and otherwise general. Assume that $(h+1)(d-2)$ is odd. If either

1. (i) $d\leq \frac {5h+3}{h+1}$ , or

2. (ii) $d\leq \frac {6h+1}{h+1}$ , $h\leq 4$ , k is $C_r$ and $h+1> 2^{r+n-h-1}$ ,

then $X_5$ is unirational.

Proof. In this case, the exceptional divisor $\widetilde {E}\subset \widetilde {X}_d$ is a general divisor of bidegree $(d-2,2)$ in $\mathbb {P}^{n-h}\times \mathbb {P}^{h}$ . Note that the discriminant of the quadric bundle $\widetilde {E}\rightarrow \mathbb {P}^{n-h}$ has degree $(h+1)(d-2)$ . Hence, the claim follows from [Mas22, Theorem 1.7] and Propositions 2.7.