Symmetric tensors on the intersection of two quadrics and Lagrangian fibration

Abstract

Let $X$ be an $n$-dimensional (smooth) intersection of two quadrics, and let ${T^{\rm{*}}}X$ be its cotangent bundle. We show that the algebra of symmetric tensors on $X$ is a polynomial algebra in $n$ variables. The corresponding map ${\rm{\Phi }}:{T^{\rm{*}}}X \to {\mathbb{C}^n}$ is a Lagrangian fibration, which admits an explicit geometric description; its general fiber is a Zariski open subset of an abelian variety, which is a quotient of a hyperelliptic Jacobian by a $2$-torsion subgroup. In dimension $3$, ${\rm{\Phi }}$ is the Hitchin fibration of the moduli space of rank $2$ bundles with fixed determinant on a curve of genus $2$.

1. Introduction

Let $X\subset \mathbb{P}^{n+2}_{\mathbb{C}}$ be a smooth $n$ -dimensional complete intersection of two quadrics, with $n\geq 2$ , and let $T^*X$ be its cotangent bundle. The $\mathbb{C}$ -algebra $H^0(T^*X,\mathscr{O}_{T^*X})$ is canonically isomorphic to the algebra of symmetric tensors $H^0(X,\mathsf{S}^{\scriptscriptstyle \bullet }T_X )$ . Recall that $T^*X$ carries a canonical symplectic structure. Our main result is the following theorem:

Theorem 1.1. (a) The vector space $W:=H^0(X,\mathsf{S}^{2}T_X )$ has dimension $n$ , and the natural map $\mathsf{S}^{\scriptscriptstyle \bullet } W\rightarrow H^0(X,\mathsf{S}^{\scriptscriptstyle \bullet }T_X )$ is an isomorphism.

(b) The corresponding map, $\Phi : T^*X\rightarrow W^*\cong \mathbb{C}^n$ , is a Lagrangian fibration.

(c) When $X$ is general, the general fiber of $\Phi$ is of the form $A\smallsetminus Z$ , where $A$ is an abelian variety and $\mathrm{codim}\ Z\geq 2$ .

We will give a precise geometric description of the map $\Phi$ and of the abelian variety $A$ in Sections 4 and 5.

1.1 Comments

(1) For $n=2$ , (a) follows from Theorem 5.1 in [DOL19], while (b) and (c) are proved in [KL22]. The proof is based on the isomorphism $T_X\cong \Omega ^1_X(1)$ . The theorem also follows from the fact that $X$ is a moduli space for parabolic rank 2 bundles on $\mathbb{P}^1$ [Cas15], so $\Phi :T^*X\rightarrow \mathbb{C}^2$ is identified to the Hitchin fibration (see [BHK10]).

For $n=3$ , $X$ is isomorphic to the moduli space of vector bundles of rank 2 and fixed determinant of odd degree [New68]; again, the theorem follows from the properties of the Hitchin fibration (see Section 2). It would be interesting to have a modular interpretation of $\Phi$ for $n \geq 4$ . Note that the Hitchin map for $G$ -bundles is homogeneous quadratic only when $G$ is $\mathrm{SL}(2)$ or a product of copies of $\mathrm{SL}(2)$ , so this limits the possibilities of using it.

(2) The map $\Phi$ is an example of an algebraically completely integrable system; see Remark 5.1. There is an abundant literature on such systems; see, for instance, [A96].

A classical example, the geodesic flow on an ellipsoid, is discussed in detail in [K80]. The corresponding Lagrangian fibration takes place on the cotangent bundle of one quadric; it is not related to our $\Phi$ . However, some of the tools we use in Sections 4 and 5, in particular the variety $\mathscr{X}$ and the family of planes $\mathscr{F}$ , appear already in [K80] (with a different purpose).

(3) Such a situation is rather exceptional: Most varieties do not admit nonzero symmetric tensors (for instance, hypersurfaces of degree $\geq 3$ [HLS22]); when they do, even for varieties as simple as quadrics, the algebra of symmetric tensors is fairly complicated (see, for instance, [BLi24]). We do not have a conceptual explanation for the particularly simple behavior in our case.

(4) For $n= 2$ or $3$ , the generality assumption on $X$ in (c) is unnecessary. It seems likely that this is the case for all $n$ , but our method does not allow us to make that conclusion.

1.2 Strategy

We will first treat the case $n=3$ , which is independent of the rest of this article (Section 2). For the general case, we will develop two different approaches. In the first one we exhibit a natural $n$ -dimensional subspace $W\subset H^0(X,\mathsf{S}^2T_X)$ , from which we deduce a map $\Phi :T^*X\rightarrow W^*\cong \mathbb{C}^n$ (Section 3). We then show that $\Phi$ has the required properties, which implies (a), (b) and (c) for general $X$ (5.1). In the second approach (Section 7), we directly prove (a) for all smooth X, by realizing $X$ as a double covering of a quadric.

1.3 Notations

Throughout this article, $X$ will be a smooth complete intersection of two quadrics in $\mathbb{P}^{n+2}$ , with $n\geq 2$ . We denote by $T^*X$ its cotangent bundle and by $\mathbb{P} T^*X$ its projectivisation in the geometric sense (not in the Grothendieck sense). If $V$ is a vector space, we denote by $\mathbb{P}(V)$ the associated projective space $V\smallsetminus \{0\}/\mathbb{C}^*$ parametrising 1-dimensional subspaces of $V$ .

2. The case $\boldsymbol{{n}} \textbf{=} \textbf{3}$

In this section we show how our general results can be obtained in the case $n=3$ by interpreting $X$ as a moduli space.

As in Section 4.1 below, we associate to $X$ a genus 2 curve $C$ such that the variety of lines in $X$ is isomorphic to $JC$ . Let us fix a line bundle $N$ on $C$ of degree $1$ ; then $X$ is isomorphic to the moduli space $\mathscr{M}$ of rank 2 stable vector bundles on $C$ with determinant $N$ [New68]. The cotangent bundle $T^*\mathscr{M}$ is naturally identified with the moduli space of Higgs bundles; that is, pairs $(E,u)$ with $E\in \mathscr{M}$ and $u:E \rightarrow E\otimes K_C$ a homomorphism with $\mathrm{Tr}u=0$ . The Hitchin map $\Phi : T^*\mathscr{M}\rightarrow H^0(K_C^2)$ associates to a pair $(E,u)$ the section ${\det u}$ of $K_C^2$ . It is a Lagrangian fibration [Hit87].

Let $\omega \in H^0(K_C^2)$ . We assume in what follows that $\omega$ vanishes at $4$ distinct points. Let $C_{\omega }$ be the curve in the cotangent bundle $T^*C$ defined by $z^2=\omega$ . The projection $\pi :C_{\omega }\rightarrow C$ is a double covering branched along $\mathrm{div}(\omega )$ , and $C_{\omega }$ is a smooth curve of genus 5. Let $P$ be the Prym variety associated to $\pi$ , that is, the kernel of the norm map $\mathrm{Nm}:JC_{\omega }\rightarrow JC$ ; it is a 3-dimensional abelian variety.

Proposition 2.1. The fibre $\Phi ^{-1}(\omega )$ is isomorphic to the complement of a curve in $P$ .

Proof. Recall that the map $L\mapsto \pi _*L$ establishes a bijective correspondence between line bundles on $C_{\omega }$ and rank 2 vector bundles $E$ on $C$ endowed with a homomorphism $u:E\rightarrow E\otimes K_C$ such that $u^2=\omega \cdot \mathrm{Id}_E$ or, equivalently, $\mathrm{Tr}u=0$ and $\det u=\omega$ (see, for instance, [BNR89]). To get $(E,u)$ in $\Phi ^{-1}(\omega )$ , we have to impose $\det E= N$ and $E$ stable. Since $\det \pi _*L= \mathrm{Nm}(L)\otimes K_C^{-1}$ , the first condition means that $L$ belongs to the translate $P_N:=\mathrm{Nm}^{-1}(K_C\otimes N)$ of $P$ .

Then the vector bundle $\pi _*L$ is unstable if and only if it contains an invertible subsheaf $M$ of degree 1; this is equivalent to saying that there is a nonzero map $\pi ^*M\rightarrow L$ ; that is, $L=\pi ^*M(p)$ for some point $p\in C_{\omega }$ . The condition $L\in P_N$ means that $M^{2}(\pi (p))=K_C\otimes N$ , so $M$ is determined by $p$ up to the 2-torsion of $JC$ . Thus the locus of line bundles $L\in P_N$ such that $\pi _*L$ is unstable is a curve.

Let $\rho :C\rightarrow \mathbb{P}^1$ be the canonical double covering, with $B\subset \mathbb{P}^1$ its branch locus. Since the homomorphism $\mathsf{S}^2H^0(K_C)\rightarrow H^0(K_C^{2})$ is surjective, the divisor of $\omega$ is of the form $\rho ^*(p+q)$ , for some $p,q\in \mathbb{P}^1$ ; by assumption, we have $p\neq q$ and $p,q\notin B$ .

Proposition 2.2. Let $\Gamma$ be the double covering of $\mathbb{P}^1$ branched along $B\cup \{p,q\}$ . There is an exact sequence

\begin{equation*}\ 0\rightarrow \mathbb {Z}/2\rightarrow J\Gamma \rightarrow P\rightarrow 0\,.\end{equation*}

Proof. Let $\chi :\mathbb{P}^1\rightarrow \mathbb{P}^1$ be the double covering branched along $\{p,q\}$ . Since $\mathrm{div}(\omega ) =\rho ^*(p+q)$ , there is a cartesian diagram of double coverings

which gives rise to two commuting involutions $\sigma, \tau$ of $C_{\omega }$ , exchanging the two sheets of $\pi$ and $\xi$ , respectively. The field of rational functions on $C_{\omega }$ is

\begin{equation*}\mathbb {C}(x,y,z)\ \mbox { with }\ y^2=f(x), z^2=g(x), \end{equation*}

where $f$ and $g$ are polynomials with $\mathrm{div} f=B$ and $\mathrm{div} g=\{p,q\}$ . Then $\sigma$ and $\tau$ change the sign of $y$ and $z$ , respectively.

The involution $\sigma \tau$ is fixed-point free, so the quotient $\Gamma :=C_{\omega } /\langle{\sigma \tau } \rangle$ has genus 3; its field of functions is $\mathbb{C}(x,w)$ , with $w=yz$ and $w^2=f(x)g(x)$ . We have again a cartesian square

Let $\alpha \in J\Gamma$ . We have $\mathrm{Nm}_{\pi }\varphi ^*\alpha =\rho ^*\mathrm{Nm}_{\psi }\alpha =0$ ; hence, $\varphi ^*$ maps $J\Gamma$ into $P\subset JC_{\omega }$ . Since $\varphi$ is étale, we have $\mathrm{Ker} \varphi ^*=\mathbb{Z}/2$ ; since $\dim J\Gamma =\dim P=3$ , $\varphi ^*$ is surjective.

3. Definition of $\Phi$

Let $Y$ be a smooth degree $d$ hypersurface in $\mathbb{P}^N$ , defined by an equation $f=0$ . Recall that one associates to $f$ a section $h_f$ of $\mathsf{S}^2\Omega ^1_Y(d)$ , the hessian or second fundamental form of $f$ [GH79]: at a point $y$ of $Y$ , the intersection of $Y$ with the tangent hyperplane $H$ to $Y$ at $y$ is a hypersurface in $H$ singular at $y$ , and $h_{f}(y)$ is the degree 2 term in the Taylor expansion of $f_{|H}$ at $y$ .

Now let $X\subset \mathbb{P}^{n+r}$ be a smooth complete intersection of $r$ hypersurfaces of degree $d$ ; let

\begin{equation*}V\subset H^0(\mathbb {P}^{n+r},\mathscr {O}_{\mathbb {P}}(d))\end{equation*}

be the $r$ -dimensional subspace of degree $d$ polynomials vanishing on $X$ . By restricting $h_{f}$ , for $f\in V$ , to $X$ , we get a linear map

\begin{equation*}V\otimes \mathscr {O}_X\longrightarrow \mathsf {S}^2\Omega ^1_X(d),\end{equation*}

which gives at each point $x\in X$ a linear space of quadratic forms on the tangent space $T_{x}(X)$ . Note that when $d=2$ , the corresponding quadrics in $\mathbb{P} (T_x(X))$ can be viewed geometrically as follows: The projective space $\mathbb{P}( T_x(X))$ can be identified with the space of lines in $\mathbb{P}^{n+r}$ passing through $x$ and tangent to $X$ ; then for each $q\in V$ , the quadric defined by $h_q(x)$ parameterises the lines passing through $x$ and contained in the quadric $\{q=0\}$ .

Now we want to consider the ‘inverse’ of the quadratic form $h_f(x)$ on $T_x(X)$ ; that is, the form on $T^*_x(X)$ given in coordinates by the cofactor matrix. Intrinsically, each $f\in V$ gives a twisted symmetric morphism

\begin{equation*}h_f:T_X\longrightarrow \Omega ^1_X(d),\end{equation*}

which induces a twisted symmetric morphism on $(n-1)$ -th exterior powers, namely,

\begin{equation*}\wedge ^{n-1}h_f:\bigwedge \nolimits ^{n-1}T_X\longrightarrow \bigwedge \nolimits ^{n-1}\Omega ^1_X((n-1)d)\,.\end{equation*}

We now observe that $K_X=\mathscr{O}_X(-n-1-r+dr)$ ; hence

\begin{equation*}\bigwedge \nolimits ^{n-1}T_X\cong \Omega ^1_X(n+1-r(d-1))\ \textrm{and} \quad \bigwedge \nolimits ^{n-1}\Omega ^1_X\cong T_X(-n-1+r(d-1))\,, \end{equation*}

so $\wedge ^{n-1}h_f$ induces a symmetric morphism from $\Omega ^1_X(n+1-r(d-1))$ to $T_X((n-1)d-n-1+r(d-1))$ , hence provides a section

\begin{equation*}\wedge ^{n-1}h_f\in H^0(X,\mathsf {S}^2T_X(d(n+2r-1)-2(n+r+1))).\end{equation*}

Being locally given by the cofactor matrix, $\wedge ^{n-1}h_f$ is homogeneous of degree $n-1$ in $f$ . Hence, we have constructed a linear map

\begin{equation*} \alpha : \mathsf {S}^{n-1}V\longrightarrow H^0(X,\mathsf {S}^2T_X(d(n+2r-1)-2(n+r+1))) \quad \mbox {such that }\ \alpha (f^{n-1})=\wedge ^{n-1}h_f\,.\end{equation*}

From now on, we restrict to the case $d=2,\,r=2$ , so $X$ is the complete intersection of two quadrics, and the previous construction gives a linear map

\begin{equation*} \alpha : \mathsf {S}^{n-1}V\longrightarrow H^0(X,\mathsf {S}^2T_X)\,.\end{equation*}

Using the canonical isomorphism $H^0(T^*X,\mathscr{O}_{T^*X})= H^0(X,\mathsf{S}^{\scriptscriptstyle \bullet }T_X)$ , we deduce from $\alpha$ a morphism

\begin{equation*}\Phi : T^*X\longrightarrow \mathsf {S}^{n-1}V^*\cong \mathbb {C}^n\, .\end{equation*}

We have $\Phi (\lambda v)=\lambda ^2\Phi (v)$ for $v\in T^*X$ , $\lambda \in \mathbb{C}$ , so $\Phi$ induces a rational map

\begin{equation*}\varphi :\mathbb {P} T^*X \dashrightarrow \mathbb {P}^{n-1},\end{equation*}

whose indeterminacy locus $Z$ is the image of $\Phi ^{-1}(0)$ .

Proposition 3.1. $(1)\ \alpha$ is injective.

$(2)\ \Phi$ is surjective.

$(3)$ The image of $Z$ by the structure map $p:\mathbb{P} T^*X\rightarrow X$ is a proper subvariety of $X$ .

Proof. Let $x$ be a general point of $X$ . We claim that the base locus in $\mathbb{P} (T_x(X))$ of the pencil of quadratic forms $\{h_q(x)\}_{q\in V}$ is smooth. Indeed, this locus can be viewed as the variety $F_x$ of lines in $X$ passing through $x$ . Let $F$ be the Fano variety of lines contained in $X$ , and let

\begin{equation*}G\subset F\times X=\{(\ell, y)\,|\,y\in \ell \}\,.\end{equation*}

Then $F$ and therefore $G$ are smooth [Reid72, Theorem 2.6], hence $F_x$ , which is the fibre above $x$ of the projection $G\rightarrow X$ , is smooth since $x$ is general. It follows that in an appropriate system of coordinates $(k_1,\ldots, k_n )$ of $T_x(X)$ , the forms $\{h_q(x)\}$ can be written as

\begin{equation*}t\sum k_i^2 +\sum \alpha _ik_i^2\quad \mbox {with}\ \alpha _i \mbox { distinct in }\mathbb {C}, \ t\in \mathbb {C}\,.\end{equation*}

Then $\wedge ^{n-1}h_q(x)$ is given by the diagonal matrix with entries $\beta _i:=\displaystyle \prod _{j\neq i}(t+\alpha _j)\ (i=1,\ldots, n)$ . These polynomials in $t$ are linearly independent; hence, they generate the space of quadratic forms on $T^*_x(X)$ , which are diagonal in the basis $(k_i)$ . This linear system has dimension $n$ , so $\alpha$ is injective; it has no base point, so $\varphi$ induces a finite, surjective morphism $\mathbb{P} (T^*_x(X))\rightarrow \mathbb{P}^{n-1}$ . Thus, $\Phi$ is surjective, and $Z\cap \mathbb{P} (T^*_x(X))=\varnothing$ , which gives (2) and (3).

We want to give a geometric construction of the rational map $\varphi : \mathbb{P} T^*X\dashrightarrow \mathbb{P}^{n-1}$ . A point of $\mathbb{P} T^*X$ is a pair $(x,H)$ , where $x\in X$ and $H$ is a hyperplane in $T_x(X)$ . Restricting the pencil $\{h_q(x)\}_{q\in V}$ to $H$ gives a pencil of quadrics on $H$ , which for general $(x,H)$ contains $n-1$ singular quadrics $q_1,\ldots, q_{n-1}$ . The subset $\{q_1,\ldots, q_{n-1}\}$ of $\mathbb{P} (V)$ corresponds to a point $\varphi _{x,H}$ of $\mathbb{P}(\mathsf{S}^{n-1}V^*)$ ; namely, the hyperplane in $\mathsf{S}^{n-1}V$ spanned by $q_1^{n-1},\ldots, q_{n-1}^{n-1}$ .

Proposition 3.2. $\varphi (x,H)=\varphi _{x,H}$ .

Proof. We can assume that $x$ is general. We have seen that the restriction of $\varphi$ to $\mathbb{P} (T^*_xX)$ is the morphism given by the linear system of quadratic forms $W\cong \mathsf{S}^{n-1}V$ spanned by the forms $\wedge ^{n-1}h_q(x)$ , for $q\in V$ ; in other words, $\varphi$ maps the point $H$ of $\mathbb{P} (T^*_x(X))$ to the hyperplane of forms in $W$ vanishing at $H$ .

On the other hand, $\varphi _{x,H}$ is the hyperplane of $\mathsf{S}^{n-1}V$ spanned by the $q^{n-1}$ for those $q\in V$ such that $h_q(x)_{|H}$ is singular; this condition is equivalent to saying that the form $\wedge ^{n-1}h_q(x)$ on $T^*_xX$ vanishes at $H$ . Therefore, $\varphi _{x,H}$ is spanned by quadratic forms vanishing at $H$ , hence coincides with $\varphi (x,H)$ .

Corollary 3.3. $\mathrm{codim} Z\geq 2$ .

Proof. Suppose that $Z$ contains a component $Z_0$ of codimension 1; since $p(Z)\neq X$ , we have $Z_0=p^{-1}(p(Z_0))$ . We claim that this is impossible; in fact, $Z$ cannot contain a fibre $p^{-1}(x)$ . Indeed, its doing this would mean that for $q\in V$ , the form $h_q(x)$ is singular along all hyperplanes $H\subset T_x(X)$ ; that is, $h_q(x)$ has rank $\leq n-2$ . But the rank of $h_q(x)$ is the rank of the restriction of $q$ to the projective tangent subspace to $X$ at $x$ . Restricting a quadratic form to a hyperplane lowers its rank by up to two. Since a general $q$ in $V$ has rank $n+3$ , its restriction to a codimension 2 subspace has rank $\geq n-1$ .

4. Fibers of $\boldsymbol\varphi$

In an appropriate system of coordinates $(x_0,\ldots, x_{n+2})$ , our variety $X$ is defined by the equations $q_1=q_2=0$ , with

\begin{equation*} q_1=\sum x_i^2\, \quad q_2=\sum \mu _ix_i^2\qquad \mbox {, with }\,\mu _i\in \mathbb {C}\ \mbox { distinct.}\end{equation*}

Let $\Pi =\mathbb{P}(V)\ (\cong \mathbb{P}^1)$ be the pencil of quadrics containing $X$ . We choose a coordinate $t$ on $\Pi$ so that the quadrics of $\Pi$ are given by $tq_1-q_2=0$ . Then the singular quadrics of $\Pi$ correspond to the points $\mu _0,\ldots, \mu _{n+2}$ .

The goal of this section is to describe the general fibre of the rational map $\varphi :\mathbb{P} T^*X\dashrightarrow \mathsf{S}^{n-1}\Pi$ $(\cong \mathbb{P}^{n-1})$ . For $\lambda =(\lambda _1,\ldots, \lambda _{n-1})\in \mathsf{S}^{n-1}\Pi$ , let $C_{\mu, \lambda }$ denote the hyperelliptic curve $y^2= \prod (t-\mu _i)\prod (t-\lambda _j)$ , of genus $n$ . We will then prove the following:

Proposition 4.1. For $\lambda$ general in $\mathsf{S}^{n-1}\Pi$ , the fibre $\varphi ^{-1}(\lambda )$ is birational to the quotient of the Jacobian $JC_{\mu, \lambda }$ by the group $\Gamma :=\{\pm 1_{JC}\}\times \Gamma ^+$ , where $\Gamma ^+\cong (\mathbb{Z}/2Z)^{n-2}$ is a group of translations by $2$ -torsion elements.

4.1 Odd-dimensional intersection of 2 quadrics

We briefly recall here the results of Reid’s thesis ([Reid72]; see also [DR76]). Let $Y\subset \mathbb{P}^{2g+1}$ be a smooth intersection of 2 quadrics, and let $\Xi \ (\cong \mathbb{P}^1)$ be the pencil of quadrics containing $Y$ . Let $\Sigma \subset \Xi$ be the subset of $2g+2$ points corresponding to singular quadrics, and let $C$ be the double covering of $\Xi$ branched along $\Sigma$ ; this is a hyperelliptic curve of genus $g$ . The intermediate Jacobian $JY$ of $Y$ is isomorphic to $JC$ (as principally polarized abelian varieties). The variety $F$ of $(g-1)$ -planes contained in $Y$ is also isomorphic to $JC$ , but this isomorphism is not canonical.

In an appropriate system of coordinates, the equations of $Y$ are of the form

\begin{equation*}\sum x_i^2=\sum \alpha _ix_i^2=0,\qquad \mbox {with } \alpha _i\in \mathbb {C} \mbox { distinct;}\end{equation*}

then $\Sigma =\{\alpha _1,\ldots, \alpha _{2g+2}\}$ . The group $\Gamma :=(\mathbb{Z}/2\mathbb{Z})^{2g+1}$ acts on $Y$ (hence also on $F$ ) by changing the signs of the coordinates. Let $\Gamma ^+\subset \Gamma$ be the subgroup of elements that change an even number of coordinates. Choose an element $\gamma \in \Gamma \smallsetminus \Gamma ^+$ ; there is an isomorphism $F\stackrel{\sim }{\longrightarrow } JC$ such that $\gamma$ corresponds to (-1_JC). Then the image of $\Gamma ^+$ in $\mathrm{Aut}(JC)$ is the group $T_2$ of translations by $2$ -torsion elements of $JC$ , and the image of $\Gamma$ is $T_2\times \{\pm 1_{JC}\}$ [DR76, Lemma 4.5].

4.2 An auxiliary construction

We consider the projective space $\mathbb{P}^{2n+1}$ equipped with the system of homogeneous coordinates

\begin{equation*}x_0,\ldots, x_{n+2};\,y_1,\ldots, \,y_{n-1}\end{equation*}

and the affine space $\mathbb{A}^{n-1}$ equipped with the affine coordinates $\lambda _1,\ldots, \lambda _{n-1}$ . Let

\begin{equation*}\mathscr {X}\subset \mathbb {P}^{2n+1}\times \mathbb {A}^{n-1}\end{equation*}

be the complete intersection of the two quadrics with equations

\begin{equation*}Q_1=Q_2=0\quad \mbox {with}\quad Q_1=\sum _{i=0}^{n+2}x_i^2+\sum _{j=1}^{n-1}y_j^2\quad, \quad Q_2=\sum _{i=0}^{n+2}\mu _ix_i^2+\sum _{j=1}^{n-1}\lambda _jy_j^2\, .\end{equation*}

The second projection, $\mathscr{X}\rightarrow \mathbb{A}^{n-1},$ gives a family of complete intersections of two quadrics $\mathscr{X}_\lambda$ of dimension $2n-1$ parameterised by $ \mathbb{A}^{n-1}$ . Note that $X$ is the intersection of $\mathscr{X}$ with the subspace $\mathbb{P}^{n+2}\subset \mathbb{P}^{2n+1}$ defined by $y_1=\ldots =y_{n-1}=0$ .

Let $p: \mathscr{F}\rightarrow \mathbb{A}^{n-1}$ be the family of $(n-1)$ -planes contained in the $\mathscr{X}_{\lambda }$ ; that is

\begin{equation*}\mathscr {F}=\{(P,\lambda )\,|\,\lambda \in \mathbb {A}^{n-1}, \ P\ (n-1)\mbox {-plane}\subset \mathscr {X}_{\lambda }\}\,. \end{equation*}

For $\lambda$ general, the fibre $\mathscr{F}_{\lambda }$ is isomorphic to the Jacobian of the hyperelliptic curve $C_{\mu, \lambda }$ (4.1).

Let $(P,\lambda )$ be a general point of $\mathscr{F}$ . Then $P\,\cap \, \mathbb{P}^{n+2}$ is a point $x$ of $X$ . Let $\pi :\mathbb{P}^{2n+1}\dashrightarrow \mathbb{P}^{n+2}$ be the projection $(x_i,y_j)\mapsto (x_i)$ . Since the $\pi_*$ differentials of $Q_i$ and $q_i$ coincide at $x$ , the differential $\pi$ _* maps $T_x(P)\subset T_x(\mathscr{X})$ into $T_x(X)$ . Since $P$ is general, $\pi _*T_x(P)$ is a hyperplane in $T_x(X)$ ; this will follow from the proof of Proposition 4.2, (1) below, where we explicitly construct pairs $(P,\lambda )$ with this property.

Therefore, we have a rational map

\begin{equation*}\psi : \mathscr {F}\dashrightarrow \mathbb {P} T^*X\,\quad (P,\lambda )\mapsto (x=P\cap \mathbb {P}^{n+2}\,, \ \pi _*T_x(P))\,.\end{equation*}

The symmetric group $\mathfrak{S}_{n-1}$ acts on $\mathbb{P}^{2n+1}$ by permuting the $y_j$ and acts on the group $(\mathbb{Z}/2\mathbb{Z})^{n-1}$ by changing their signs; this gives an action of the semi-direct product $G:=(\mathbb{Z}/2\mathbb{Z})^{n-1}\rtimes \mathfrak{S}_{n-1}$ . We make $G$ act on $\mathbb{A}^{n-1}$ through its quotient $\mathfrak{S}_{n-1}$ , by permutation of the $\lambda _i$ . This induces an action of $G$ on $\mathscr{X}$ and therefore on $\mathscr{F}$ , which is compatible via $p$ with the action on the base. The map $\psi$ is invariant under this action; hence, it factors through the quotient $\mathscr{F}/G$ . By passing to the quotient, we get a map $p^\sharp : \mathscr{F}/G\rightarrow \mathbb{A}^{n-1} /\mathfrak{S}_{n-1}$ .

Proposition 4.2. $(1)\ \psi$ induces a birational map $\psi ^{\sharp }:\mathscr{F}/G\dashrightarrow \mathbb{P} T^*X$ .

$(2)$ There is a commutative diagram

where $p^{\sharp }$ is deduced from $p$ and where $\sigma$ is the isomorphism given by symmetric functions.

Proof. (1) Let $(x,H)\in \mathbb{P} T^*X$ ; we want to describe the pairs $(P,\lambda )$ such that $P\cap \mathbb{P}^{n+2}=\{x\}$ and $\pi _*T_x(P)=H$ . The latter condition says that via the decomposition

\begin{equation*}T_x(\mathbb {P}^{2n+1})=T_x(\mathbb {P}^{n+2})\oplus \mathrm {Ker}\,\pi _*\,,\end{equation*}

$T_x(P)$ identifies with the graph of a linear map

\begin{equation*}\alpha : H\rightarrow \mathrm {Ker}\, \pi _*\, .\end{equation*}

Using the basis $(\frac{\partial }{\partial y_1},\ldots, \frac{\partial }{\partial y_{n-1}} )$ of $\mathrm{Ker}\, \pi _*$ , we have $\alpha =(\alpha _1,\ldots, \alpha _{n-1})$ , where the $\alpha _i$ are linear forms on $H$ . The condition $P\subset \mathscr{X}_{\lambda }$ implies that the hessians $h_{Q_1}(x)$ and $h_{Q_2}(x)$ vanish on $T_x(P)$ , which gives

(1) \begin{equation} h_{q_1}(x)_{\mid H}=-\sum _i \alpha _i^2\, \quad h_{q_2}(x)_{\mid H}=-\sum _i\lambda _i\alpha _i^2\,. \end{equation}

This is a simultaneous diagonalisation of the quadratic forms $h_{q_1}(x)_{\mid H}$ and $h_{q_2}(x)_{\mid H}$ ; when they are in general position, this determines the $\lambda _i$ up to permutation and the $\alpha _i$ up to sign and permutation, which proves (1).

(2) Let $(P,\lambda )\in \mathscr{F}$ , and let $(x,H):=\psi (P,\lambda )$ . According to Proposition 3.2, $\varphi (x,H)$ is given by the $(n-1)$ -uple of quadrics $ q\in \Pi$ such that the form $h_{q}(x)_{\mid H}$ is singular. Using $(\alpha _1,\ldots, \alpha _{n-1})$ as coordinates on $H$ , we see from (1) that this $(n-1)$ -uple is given by $(\lambda _1,\ldots, \lambda _{n-1})$ , which proves (2).

4.3 Proof of Proposition 4.1.

Let $\lambda$ be a general element of $\mathbb{A}^{n-1}$ . Let us denote by $\Gamma$ the subgroup $(\mathbb{Z}/2\mathbb{Z})^{n-1}$ of $G$ . From Proposition 4.2 and the cartesian diagram

we see that the fibre $\varphi ^{-1}(\lambda )$ is birational to the quotient $\mathscr{F}_{\lambda }/\Gamma$ . By (4.1) there is an isomorphism $\mathscr{F}_{\lambda } \stackrel{\sim }{\longrightarrow } JC_{\mu, \lambda }$ such that $\Gamma$ acts on $JC_{\mu, \lambda }$ as $\{\pm 1_{J}\} \times \Gamma ^+$ , where $\Gamma ^+$ is a group of translations by $2$ -torsion elements. This proves the proposition.

5. Fibres of $\Phi$

5.1 Results

We keep the settings of the previous section. Recall that our parameter $\lambda$ lives in $\mathbb{A}^{n-1}\subset \mathsf{S}^{n-1}\Pi \cong \mathbb{P}^{n-1}$ . For $\lambda$ in $\mathbb{A}^{n-1}$ , we denote by $\tilde{\lambda }$ a lift of $\lambda$ in $\mathbb{C}^n$ for the quotient map $\mathbb{C}^n\smallsetminus \{0\}\rightarrow \mathbb{P}^{n-1}$ .

Theorem 5.1. Assume that $X$ is general. For $\lambda \in \mathbb{A}^{n-1}$ general, the fibre $\Phi ^{-1}(\tilde{\lambda } )$ is isomorphic to $A\smallsetminus Z$ , where:

$\bullet \ A$ is the abelian variety quotient of $JC_{\mu, \lambda }$ by a $2$ -torsion subgroup, isomorphic to $(\mathbb{Z}/2\mathbb{Z})^{n-2}\,$ ;

$\bullet \ Z$ is a closed subvariety of codimension $\geq 2$ in $A$ .

Corollary 5.2. For every smooth complete intersection of two quadrics $X\subset \mathbb{P}^{n+2}$ , the fibration $\Phi :T^*X\rightarrow \mathbb{C}^n$ is Lagrangian.

Proof. Assume first that $X$ is general. The symplectic form on $T^*X$ is $d\eta$ , where $\eta$ is the Liouville form. By Theorem 5.1 and Hartogs’ principle, the pull-back of $\eta$ to a general fibre of $\Phi$ is the restriction of a 1-form on an abelian variety, hence is closed. This implies the result.

Let $p:\mathscr{X}\rightarrow B$ be a complete family of smooth intersection of two quadrics in $\mathbb{P}^{n+2}$ . The constructions of §3 can be globalised over $B$ : we have a rank 2 vector bundle $\mathscr{V}$ over $B$ whose fibre at a point $b\in B$ is the space of quadratic forms vanishing on $\mathscr{X}_b$ . We get a homomorphism $\mathsf{S}^{n-1}\mathscr{V}\rightarrow p_*T_{\mathscr{X}/B}$ , which thus gives rise to a morphism $\boldsymbol{\Phi }:T^*(\mathscr{X}/B)\rightarrow \mathsf{S}^{n-1}\mathscr{V}^*$ over $B$ which induces over each point $b\in B$ our map $\Phi$ . There is a natural Liouville form $\boldsymbol{\eta }$ on $T^*(\mathscr{X}/B)$ : Since $d\boldsymbol{\eta }$ vanishes on a general fibre of $\boldsymbol{\Phi }$ , it vanishes on all fibres.

Corollary 5.3. Assume that $X$ is general. The multiplication map $\mathsf{S}^{\scriptscriptstyle \bullet } H^0(X,\mathsf{S}^2T_X)\rightarrow H^0(X,\mathsf{S}^{\scriptscriptstyle \bullet }T_X)$ is an isomorphism.

(We will give in Section 7 a proof that is valid with no generality assumption.)

Proof. Theorem 5.1 implies that every function on a general fibre of $\Phi$ is constant; hence, the pull-back $\Phi ^*:H^0(\mathbb{C}^n,\mathscr{O}_{\mathbb{C}^n})\rightarrow H^0(T^*X,\mathscr{O}_{T^*X})$ is an isomorphism. The right-hand space is canonically isomorphic to $H^0(X,\mathsf{S}^{\scriptscriptstyle \bullet }T_X)$ ; hence, we get an algebra isomorphism $\mathbb{C}[t_1,\ldots, t_n]\stackrel{\sim }{\longrightarrow } H^0(X,\mathsf{S}^{\scriptscriptstyle \bullet }T_X)$ . By construction, the $t_i$ are mapped to elements of $H^0(X,\mathsf{S}^{2}T_X)$ , so the Corollary follows.

Remark 5.4. Let $V_1,\ldots, V_n$ be the Hamiltonian vector fields on $T^*X$ that are associated to the components of $\Phi$ . For $\lambda$ general in $\mathbb{C}^n$ , let us identify $\Phi ^{-1}(\lambda )$ to $A\smallsetminus Z$ as in the theorem. Then by Hartogs’ principle the $V_i$ linearise on $A$ ; that is, they extend to a basis of $H^0(A,T_A)$ . In principle, this allows to write explicit solutions of the Hamilton equations for $\Phi _i$ in terms of theta functions.

5.2 Proof of the theorem: lemmas

We fix a general point $\lambda \in \mathbb{A}^{n-1}$ . We denote by $\mathscr{F}^{\mathrm{o}}$ the open subset of $\mathscr{F}$ where the rational map $\psi$ is well-defined and denote by $\mathscr{F}^{\mathrm{o}}_{\lambda }$ its intersection with the fibre $\mathscr{F}_{\lambda }$ . Since $\lambda$ is general, the complement of $\mathscr{F}^{\mathrm{o}}_{\lambda }$ in $\mathscr{F}_{\lambda }$ has codimension $\geq 2$ . The rational map $\psi$ induces a morphism $\psi ^{\mathrm{o}}: \mathscr{F}^{\mathrm{o}}\rightarrow \mathbb{P} T^*X$ ; we denote by $\psi ^{\mathrm{o}}_{\lambda }$ its restriction to $\mathscr{F}_{\lambda }^{\mathrm{o}}$ . Let $Z\subset \mathbb{P} T^*X$ be the indeterminacy locus of $\varphi$ (§ 3), and let $\mathscr{F}_{\lambda }^{\mathrm{bad} }:= (\psi ^{\mathrm{o}}_{\lambda })^{-1}(Z)\subset \mathscr{F}^{\mathrm{o}}_{\lambda }$ .

Proposition 5.5. $\mathscr{F}_{\lambda }^{\mathrm{bad} }$ has codimension $\geq 2$ in $\mathscr{F}_{\lambda }$ .

We postpone the proof of Proposition 5.5 to the next section; here we show how it implies Theorem 5.1.

Let $0_X\subset T^*X$ be the zero section, and let $q:T^*X\smallsetminus 0_X\rightarrow \mathbb{P} T^*X$ be the quotient map. Let $\varphi ^{\mathrm{o}}: \mathbb{P} T^*X\smallsetminus Z\rightarrow \mathbb{P}^{n-1}$ be the morphism induced by $\varphi$ . We thus have $\ q(\Phi ^{-1}(\tilde{\lambda } ))= (\varphi ^{\mathrm{o}})^{-1}(\lambda )$ , and the restriction

\begin{equation*}q_{\lambda }: \Phi ^{-1}(\tilde {\lambda } )\rightarrow (\varphi ^{\mathrm {o}})^{-1}(\lambda )\end{equation*}

is an étale double cover, with Galois involution $\iota$ induced by $(-1_{T^*X})$ .

We put $\mathscr{F}_{\lambda }^{\mathrm{oo}}:= \mathscr{F}_{\lambda }^{\mathrm{o}}\smallsetminus \mathscr{F}_{\lambda }^{\mathrm{bad} }$ and consider the restriction

\begin{equation*}\psi _{\lambda } ^{\mathrm {o}}:\mathscr {F}_{\lambda }^{\mathrm {oo}}\rightarrow (\varphi ^{\mathrm {o}})^{-1}(\lambda )\quad \mbox {of } \ \psi ^{\mathrm {o}}\,.\end{equation*}

Lemma 5.6. The fibre $\Phi ^{-1}(\tilde{\lambda })$ is Lagrangian, and has a trivial tangent bundle.

Proof. The étale double cover $q_{\lambda }$ induces by fibred product an étale double cover

\begin{equation*}\pi :\widetilde {\mathscr {F}}_{\lambda }^{\mathrm {oo}}\rightarrow \mathscr {F}_{\lambda }^{\mathrm {oo}}\end{equation*}

such that $\psi ^{\mathrm{o}}_{\lambda }$ lifts to a morphism $\tilde{\psi }^{\mathrm{o}}_{\lambda }: \widetilde{\mathscr{F}}_{\lambda }^{\mathrm{oo}}\rightarrow \Phi ^{-1}(\tilde{\lambda } )$ .

By Proposition 5.5, the complement of $\mathscr{F}_{\lambda }^{\mathrm{oo}}$ in $\mathscr{F}_{\lambda }$ has codimension $\geq 2$ , so $\pi$ extends to an étale double cover $\widetilde{\mathscr{F}}_{\lambda }\rightarrow \mathscr{F}_{\lambda }$ , where $\widetilde{\mathscr{F}}_{\lambda }$ is an abelian variety or the disjoint union of two abelian varieties. The morphism $\tilde{\psi }^{\mathrm{o}}_{\lambda }: \widetilde{\mathscr{F}}_{\lambda }^{\mathrm{oo}}\rightarrow \Phi ^{-1}(\tilde{\lambda } )$ is generically of maximal rank. Again by Proposition 5.5, the holomorphic 1-forms on $\widetilde{\mathscr{F}}_{\lambda }^{\mathrm{oo}}$ are closed; hence by pull-back, the same holds for the holomorphic 1-forms on $\Phi ^{-1}(\tilde{\lambda } )$ . As in the proof of Corollary 5.2, this implies that $\Phi ^{-1}(\tilde{\lambda } )$ is Lagrangian. The second assertion is a basic property of Lagrangian fibres.

Lemma 5.7 The morphism $\psi ^{\mathrm{o}}_{\lambda }$ lifts to a morphism $\tilde{\psi }^{\mathrm{o}}_{\lambda }:\mathscr{F}^{\mathrm{oo}}_{\lambda }\rightarrow \Phi ^{-1}(\tilde{\lambda } )$ .

Proof. It suffices to show that the double covering $\pi :\widetilde{\mathscr{F}}_{\lambda }^{\mathrm{oo}}\rightarrow \mathscr{F}_{\lambda }^{\mathrm{oo}}$ splits.

Assuming the contrary, $\widetilde{\mathscr{F}}_{\lambda }$ is an abelian variety. By Lemma 5.6 $H^0(\Phi ^{-1}(\tilde{\lambda }), \Omega ^1)$ has dimension $n$ . It follows that the pull-back $(\tilde{\psi }^{\mathrm{o}}_{\lambda })^*: H^0(\Phi ^{-1}(\tilde{\lambda }), \Omega ^1) \rightarrow H^0(\widetilde{\mathscr{F}}_{\lambda }^{\mathrm{oo}},\Omega ^1)$ is bijective. Since the Galois involution of the double covering $\pi$ acts trivially on holomorphic 1-forms, the same holds for the Galois involution $\iota$ of the double covering $q_{\lambda }: \Phi ^{-1}(\tilde{\lambda } )\rightarrow (\varphi ^{\mathrm{o}})^{-1}(\lambda )$ .

Now we observe that the 1-forms on $\Phi ^{-1}(\tilde{\lambda })$ are ‘pure’; that is, they extend to any smooth projective compactification of $\Phi ^{-1}(\tilde{\lambda } )$ . This follows from the fact that this holds after pull-back to $\widetilde{\mathscr{F}}_{\lambda }^{\mathrm{oo}}$ . But the quotient $\Phi ^{-1}(\tilde{\lambda } )/\iota$ is isomorphic to a Zariski open subset of $\varphi ^{-1}(\lambda )$ , which, by Proposition 4.1, has no nonzero holomorphic 1-forms, so any Zariski open subset has no nonzero closed pure holomorphic 1-forms. This contradiction proves the lemma.

5.3 Proof of Theorem 5.1

Lemma 5.7 gives a factorisation,

\begin{equation*}\psi ^{\mathrm {o}}_{\lambda }: \mathscr {F}^{\mathrm {oo}}_{\lambda } \xrightarrow {\ \tilde {\psi }^{\mathrm {o}}_{\lambda } \ } \Phi ^{-1}(\tilde {\lambda } )\xrightarrow {\ q_{\lambda }\ } (\varphi ^{\mathrm {o}})^{-1}(\lambda )\, .\end{equation*}

By Proposition 4.1, $\psi ^{\mathrm{o}}_{\lambda }$ induces a birational morphism,

\begin{equation*}\psi ^{\mathrm {o}}_{\lambda, \Gamma }: \mathscr {F}^{\mathrm {oo}}_{\lambda }/\Gamma \longrightarrow (\varphi ^{\mathrm {o}})^{-1}(\lambda )\,\end{equation*}

it follows that for some subgroup $\Gamma '\subset \Gamma$ of index $2$ , the morphism $\tilde{\psi _{\lambda }^{\mathrm{o}}}:\mathscr{F}_\lambda ^{\mathrm{oo}}\rightarrow \Phi ^{-1}(\tilde{\lambda } )$ factors through a birational morphism,

\begin{equation*}\tilde {\psi }^{\mathrm {o}}_{\lambda, \Gamma '}: \mathscr {F}^{\mathrm {oo}}_{\lambda }/\Gamma '\longrightarrow \Phi ^{-1}(\tilde {\lambda } )\,.\end{equation*}

By Lemma 5.6, the cotangent bundle of $\Phi ^{-1}(\tilde{\lambda } )$ is trivial. Therefore, the cotangent bundle of $\mathscr{F}_{\lambda }^{\mathrm{oo}}/\Gamma '$ is generically generated by its global sections. This implies that $\Gamma '$ acts trivially on holomorphic 1-forms and, hence, is the subgroup $\Gamma ^{+}$ of $\Gamma$ generated by translations, isomorphic to $(\mathbb{Z}/2\mathbb{Z})^{n-2}$ ; thus $\mathscr{F}_{\lambda }/\Gamma '$ is an abelian variety $A$ .

To simplify notation, we write $A^{\mathrm{o}}:=\mathscr{F}^{\mathrm{oo}}_{\lambda }/\Gamma '$ and $u:=\tilde{\psi }^{\mathrm{o}}_{\lambda, \Gamma '}$ . The rational map $u^{-1}: \Phi ^{-1}(\tilde{\lambda } )\dashrightarrow A$ is everywhere defined (e.g. [BL92, Theorem 4.9.4]), so we have two morphisms

\begin{equation*}A^{\mathrm {o}} \xrightarrow {\ u\ } \Phi ^{-1}(\tilde {\lambda } ) \xrightarrow {\ u^{-1}\ } A\end{equation*}

whose composition is the inclusion $A^{\mathrm{o}}\hookrightarrow A$ . Since the tangent bundles of $A$ and $\Phi ^{-1}(\tilde{\lambda } )$ are trivial, the determinant of $Tu: T_{A^{\mathrm{o}}}\rightarrow u^*T _{\Phi ^{-1}(\tilde{\lambda } )}$ is a function on $A^{\mathrm{o}}$ , hence constant by Proposition 5.5. Therefore, $u$ is étale and birational, hence an open embedding. This implies that every function on $\Phi ^{-1}(\widetilde{\lambda })$ is constant (because its restriction to $A^{\mathrm{o}}$ is constant). Then the previous argument shows that $u^{-1}$ is also an open embedding, hence $\Phi ^{-1}(\tilde{\lambda } )$ is isomorphic to an open subset of $A$ containing $A^{\mathrm{o}}$ . This proves the theorem.

6. Proof of Proposition 5.5

We keep the notations of Section 4.2. Recall that we have coordinates $(x_0,\ldots, x_{n+2};y_1,\ldots, y_{n-1})$ on $\mathbb{P}^{2n+1}$ and subspaces $\mathbb{P}^{n+2}$ and $\mathbb{P}^{n-2}$ in $\mathbb{P}^{2n+1}$ defined by $y=0$ and $x=0$ .

Let $q_1(x)=q_2(x)=0$ be the equations defining $X$ in $\mathbb{P}^{n+2}$ , and let $R$ be the vector space of quadratic forms in $y=(y_1,\ldots, y_{n-1})$ . We define an extended family $\mathscr{X}^{e}\subset \mathbb{P}^{2n+1}\times R^2$ as

\begin{equation*}\mathscr {X}^{e}=\{\bigl ((x,y);(r_1,r_2)\bigr )\in \mathbb {P}^{2n+1}\times R^2\ |\ q_1(x)+r_1(y)=q_2(x)+r_2(y)=0\}\, .\end{equation*}

The fibre $\mathscr{X}^{e}_{r}$ at a point $r=(r_1,r_2)$ of $R^2$ is the intersection in $\mathbb{P}^{2n+1}$ of the two quadrics $q_1(x)+r_1(y)= q_2(x)+r_2(y)=0$ . Let $\mathbb{G}$ be the Grassmannian of $(n-1)$ -planes in $\mathbb{P}^{2n+1}$ ; we define as before

\begin{equation*}\mathscr {F}^{e}:=\{(P,r)\in \mathbb {G}\times R^2\,|\,P \subset \mathscr {X}^{e}_{r}\}\end{equation*}

and the extended rational map $\psi ^{e}:\mathscr{F}^{e}\dashrightarrow \mathbb{P} T^*X$ , which maps a general $P\subset \mathscr{X}^{e}_r$ to the pair $(x,H)$ , with $\{x\}=P\cap \mathbb{P}^{n+2}$ and $H=\pi _*T_x(P)$ .

We observe that a general pair $r=(r_1,r_2)$ of $R^2$ is simultaneously diagonalisable, so the restriction of $\psi ^{e}$ to $\mathscr{F}^{e}_r$ coincides, for an appropriate choice of the coordinates $(y_i)$ , with the map $\psi _{\lambda }$ that we want to study.

Proposition 6.1. Assume that $X$ is general.

$1.$ Let $\Gamma \subset \mathscr{F}^{e}$ be the locus of points $(P,r)$ such that either $\dim P\cap \mathbb{P}^{n+2}\gt 0$ , or $P\cap \mathbb{P}^{n-2}\neq \varnothing$ . Then $\Gamma$ has codimension $\geq 2$ in $\mathscr{F}^{e}$ .

$2.$ There exists no divisor in $\mathscr{F}^{e}\smallsetminus \Gamma$ that dominates $R^2$ and that is mapped to the base locus $Z\subset \mathbb{P} T^*X$ by $\psi _e$ .

We claim that this implies Proposition 5.5. Indeed, as just explained above, it suffices to prove the analogue of Proposition 5.5 for $\psi ^e$ . Next, it is clear that the indeterminacy locus of $\psi ^e$ is contained in $\Gamma$ , so $\psi ^e$ is well-defined on $\mathcal{F}^e\smallsetminus \Gamma$ . By Proposition 6.1, (1), it now suffices to prove the analogue of Proposition 5.5 for the restriction of $\psi ^e$ to $\mathcal{F}^e\smallsetminus \Gamma$ . This is exactly the statement of Proposition 6.1, (2).

Proof of Proposition 6.1 : (1) Let $\mathcal{Q}$ be the vector space of quadratic forms on $\mathbb{P}^{2n+1}$ of the form $q(x)+r(y)$ for some quadratic forms $q$ and $r$ . For each pair of integers $(k,l)$ with $k\geq 0$ , $l\geq -1$ , let $\mathbb{G}_{k,l}$ be the locally closed subvariety of $(n-1)$ -planes $P\in \mathbb{G}$ such that

\begin{equation*}\dim (P\cap \mathbb {P}^{n+2})=k\, \quad \dim (P\cap \mathbb {P}^{n-2})=l\,.\end{equation*}

(We put, by convention, $l=-1$ if $P\cap \mathbb{P}^{n-2}=\varnothing$ .) Let

\begin{equation*} \mathscr {F}^{\mathcal {Q}}:= \{(P, (Q_1,Q_2))\in \mathbb {G}\times \mathcal {Q}^2\ |\ Q_{1|P}=Q_{2|P}=0\}\, \end{equation*}

and

\begin{equation*}\mathscr {F}^{\mathcal {Q}}_{k,l}:= \mathscr {F}^{\mathcal {Q}}\cap (\mathbb {G}_{k,l}\times \mathcal {Q}^2)\,.\end{equation*}

The general fibre of the projection $\mathscr{F}^{\mathcal{Q}}\rightarrow \mathcal{Q}^2$ is an abelian variety, and we recover $\mathscr{F}^{e}$ by restricting $\mathscr{F}^{\mathcal{Q}}$ to pairs of quadratic forms of the form $(q_{1}(x)+r_{1}(y), q_{2}(x)+r_{2}(y))$ , with $(q_1(x),q_2(x))$ fixed. Because we assume $X$ general, the pair $(q_1(x),q_2(x))$ is general in $\mathcal{Q}^2$ . It thus suffices to prove the result for the larger family $\mathscr{F}^{\mathcal{Q}}$ ; that is, to show that $\mathscr{F}_{k,l}^{\mathcal{Q}}$ has codimension $\geq 2$ in $\mathscr{F}^{\mathcal{Q}}$ .

This is done by a dimension count. For $P\in \mathbb{G}$ , let $\varphi _P$ be the restriction map $\mathcal{Q}\rightarrow H^0(P,\mathscr{O}_{P}(2))$ . The fibre of the projection $\mathscr{F}^{\mathcal{Q}}\rightarrow \mathbb{G}$ is the vector space $(\mathrm{Ker} \varphi _P)^{\oplus 2}$ . For $P$ general, $\varphi _P$ is surjective: This is the case, for instance, if $P$ is contained in the $(n+2)$ -plane in $\mathbb{P}^{2n+1}$ defined by $y_i=x_i\ (i=1,\ldots, n-1)$ . However, $\varphi _P$ is not surjective for $P\in \mathbb{G}_{k,l}$ because the forms $r(y)_{|P}$ are singular along $P\cap \mathbb{P}^{n+2}$ and the forms $q(x)_{|P}$ are singular along $P\cap \mathbb{P}^{n-2}$ ; this implies that the subspaces $P\cap \mathbb{P}^{n+2}$ and $P\cap \mathbb{P}^{n-2}$ are apolar for all forms in $\mathrm{Im} \varphi _P$ . Therefore, the corank of $\varphi _P$ is $\geq (k+1)(l+1)$ , and there is equality when $P$ is contained in the subspace defined by $x_0=\ldots =x_{n+1-k}=y_1=\ldots =y_{n-2-l}=0$ , hence for $P$ general in $\mathbb{G}_{k,l}$ . Thus our assertion follows from:

\begin{align*} \mathrm{codim}(\mathscr{F}^{\mathcal{Q}}_{k,l},\mathscr{F}^{\mathcal{Q}})&=\mathrm{codim}(\mathbb{G}_{k,l},\mathbb{G})-2(k+1)(l+1) \\ &= k(k+1)+(l+1)(l+4)-2(k+1)(l+1)\\ &=(k-l)(k-l-1)+2(l+1)\\ &\geq 2\quad \mbox{if }\ k\geq 1\ \mbox{ or }\ l\geq 0\,. \end{align*}

(2) The base locus $Z\subset \mathbb{P} T^*X$ has codimension $\geq 2$ (Corollary 3.3). Note that $\psi ^{e}$ is well-defined in $\mathscr{F}^{e}\smallsetminus \Gamma$ . If $\mathscr{D}$ is a codimension $1$ subvariety in $\mathscr{F}^{e}\smallsetminus \Gamma$ , with $\psi ^{e}(\mathscr{D})\subset Z$ , the map $\psi ^{e}$ does not have maximal rank along $\mathscr{D}$ . This contradicts the following lemma:

Lemma 6.2. $\psi ^{e}$ has maximal rank on $\mathscr{F}^{e}\smallsetminus \Gamma$ .

Proof. Let $(x,H)$ be a point of $\mathbb{P} T^*X$ ; we view $H$ as a hyperplane in the projective tangent space to $x$ at $X$ . The fibre of $\psi ^{e}: \mathscr{F}^{e}\smallsetminus \Gamma \rightarrow \mathbb{P} T^*X$ at $(x,H)$ is the locus

(2) \begin{equation} (\psi ^{e})^{-1}(x,H)=\{(P,r_1,r_2)\in \mathbb{G}\times R^2\ |\ P\cap \mathbb{P}^{n+2} =\{x\}\,, \ P\cap \mathbb{P}^{n-2}=\varnothing \,, \ \pi (P)=H, \end{equation}

(3) \begin{equation} (q_i+r_i)_{|P} =0\quad (i=1,2)\} \,. \end{equation}

The equations (2) define a smooth, locally closed subvariety $\mathbb{G}_{x,H}$ of $\mathbb{G}$ . Let $P\in \mathbb{G}_{x,H}$ , and let $\chi _P: R\rightarrow H^0(P,\mathscr{O}_P(2))$ be the restriction map. We will show below that the image of $\chi _P$ is the space of quadratic forms on $P$ that are singular at $x$ . Since the forms $q_{i|P}$ are singular at $x$ , this implies that the solutions of (3) form an affine space over $(\mathrm{Ker} \chi _P)^{\oplus 2}$ . Therefore, $(\psi ^{e})^{-1}(x,H)$ admits an affine fibration over $\mathbb{G}_{x,H}$ , hence is smooth.

Clearly the quadrics in $\mathrm{Im} \chi _P$ are singular at $x$ . To prove the opposite inclusion, choose the coordinates $(x_i)$ so that $x=(1,0,\ldots, 0)$ . Since $P\cap \mathbb{P}^{n+2}=\{x\}$ , there exist linear forms $\ell _1,\ldots, \ell _{n+2}$ in the $y_j$ so that $P$ is defined by $x_i=\ell _{i}(y)$ for $i=1,\ldots, n+2$ . Then a quadratic form on $\mathbb{P}^{2n+1}$ singular at $x$ can be written as a form in $x_1,\ldots, x_{n+2} ;y_1,\ldots, y_{n-1}$ ; hence, its restriction to $P$ is in $\mathrm{Im} \chi _P$ . This proves the lemma and, hence, also the proposition.

7. Symmetric tensors: second approach

7.1 The cotangent bundle of a smooth quadric

We consider a smooth quadric $Q\subset \mathbb{P}^{n+1}$ defined by an equation $q=0$ . Its cotangent bundle $\mathbb{P} T^*Q$ parameterises pairs $(x,P)$ with $x\in Q$ and $P$ a $(n-1)$ -plane tangent to $Q$ at $x$ . Thus, we get a morphism $\gamma$ from $\mathbb{P} T^*Q$ to the Grassmannian $\mathbb{G}$ of $(n-1)$ -planes in $\mathbb{P}^{n+1}$ , which is the morphism defined by the linear system $\lvert{\mathscr{O}_{\mathbb{P} T^*Q}(1)}\rvert$ . It is birational onto its image, but contracts the subvariety $\mathscr{C}\subset \mathbb{P} T^*Q$ that consists of pairs $(x,P)$ , such that $P$ is tangent to $Q$ along a line $\ell \subset Q$ with $x\in \ell$ ; then $\gamma ^{-1}(P)$ consists of the pairs $(x,P)$ with $x\in \ell$ .

Let $h_q\in H^0(Q,\mathsf{S}^2 \Omega ^1 _Q(2))$ be the hessian form of $q$ (§3). Choosing coordinates $(x_i)$ such that $q(x)=\sum x_i^2$ , we have $h_q=\sum (dx_i)^2$ (note that this is, up to a scalar, the unique element of $H^0(Q,\mathsf{S}^2 \Omega ^1 _Q(2))$ invariant under $\mathrm{Aut}(Q)$ ). Then $h_q(x)$ is non-degenerate at each point $x$ of $Q$ , so $h_q$ induces an isomorphism $\Omega ^1 _Q(1)\stackrel{\sim }{\longrightarrow } T_Q(-1)$ , hence also $\mathsf{S}^2 \Omega ^1 _Q(2)\stackrel{\sim }{\longrightarrow } \mathsf{S}^2 T_Q(-2)$ . The image in $H^0(Q,\mathsf{S}^2 T_Q(-2))$ of $h_q$ by this isomorphism is $h'_q=\sum \partial _j^2$ . We will view $h'_q$ as an element of $H^0(\mathbb{P} T^*Q,\mathscr{O}_{\mathbb{P} T^*Q}(2)\otimes p^*\mathscr{O}_Q(-2))$ , where $p: \mathbb{P} T^*Q\rightarrow Q$ is the projection.

Proposition 7.1. The divisor of $h'_q$ is $\mathscr{C}$ . The projection $p_{|\mathscr{C}}:\mathscr{C}\rightarrow Q$ is a smooth quadric fibration, and $\mathscr{C}$ is a prime divisor for $n\geq 3$ .

Proof. Let $x\in Q$ ; the hyperplane tangent to $Q$ at $x$ cuts out a cone over the smooth quadric $Q_x\subset \mathbb{P}(T_x(Q))$ defined by $h_q(x)=0$ (Section 3). The isomorphism $T_x(Q)\stackrel{\sim }{\longrightarrow } T^*_x(Q)$ given by $h_q(x)$ carries $Q_x$ into the dual quadric $Q_x^*$ in $\mathbb{P}(T^*_x(Q))$ . On the other hand, a point $y\in p^{-1}(x)$ corresponds to a hyperplane $H_y\subset \mathbb{P}(T_x(Q))$ , and $y$ belongs to $\mathscr{C}$ if and only if $H_y$ is tangent to $Q_x$ ; that is, if $y\in Q_x^*$ . This proves the equality $\mathscr{C}=\mathrm{div} (h'_q)$ and thus, that the fiber of $p_{|\mathscr{C}}:\mathscr{C}\rightarrow Q$ at $x$ is $Q_x$ , which is smooth and connected if $n\geq 3$ .

Remark 7.2 The variety $\mathscr{C}$ is an example of a total dual VMRT [HLS22]. For the proof of the theorem, we will combine this tool with the birational transformation of $\mathbb{P} T^* X$ defined by a double cover. (Compare with [AH23]).

We will have to consider the following situation: Let $Q'$ be another quadric in $\mathbb{P}^{n+1}$ , such that the intersection $B:=Q\cap Q'$ is a smooth hypersurface in $Q$ . The surjection $T_{Q}\rightarrow N_{B/Q}$ gives a section of $\mathbb{P} T^*Q$ over $B$ , hence an embedding $s:B\hookrightarrow \mathbb{P} T^*Q$ .

Lemma 7.3. The image $s(B)$ is not contained in $\mathscr{C}$ .

Proof. Let $x\in B$ . The point $s(x)$ in $\mathbb{P}(T_x^*(Q))$ corresponds to the hyperplane image of $T_x(B)$ in $T_{x}(Q)$ ; we must therefore show that this hyperplane is not tangent to the quadric $Q_x:=h_q(x)$ . In terms of projective space, this means that the projective tangent space to $Q'$ at $x$ is not tangent, at a smooth point $y$ , to the cone cut out on $Q$ by the projective tangent space to $Q'$ at $x$ .

Suppose that this is the case, with $y=(y_0,\ldots, y_{n+1})$ . We can assume that $Q'$ is defined by $\sum \alpha _i x_i^2=0$ , with $\alpha _i\in \mathbb{C}$ distinct. Then the (projective) tangent space to $Q'$ at $x$ , given by $\sum (\alpha _ix_i)\xi _i=0$ , must coincide with the tangent space to $Q$ at $y$ , given by $\sum y_i\xi _i=0$ . This implies $y=(\alpha _0x_0,\ldots, \alpha _{n+1}x_{n+1})$ . Thus the point $x$ must satisfy

\begin{equation*}\sum x_i^{2}=\sum \alpha _ix_i^2=\sum \alpha _i^2 x_i^2=0\,.\end{equation*}

If these relations hold for all $x$ in $B$ , the quadric $\sum \alpha _i^2 x_i^2=0$ must belong to the pencil spanned by $Q$ and $Q'$ . This means that there exist scalars $\lambda, \mu, \nu$ such that

\begin{equation*}\lambda \alpha _i^2+\mu \alpha _i+\nu =0\quad \mbox {for all }i\,,\end{equation*}

which is impossible since the $\alpha _i$ are distinct. Therefore, there exists $x\in B$ such that $s(x)\notin \mathscr{C}$ .

7.2 Explicit description of symmetric tensors

We keep the notation of the previous sections: $X\subset \mathbb{P}=\mathbb{P}^{n+2}$ is defined by $q_1=q_2=0$ , and with

\begin{equation*} q_1 = \sum _{i=0}^{n+2} x_i^2, \qquad q_2 = \sum _{i=0}^{n+2} \mu _i x_i^2 \qquad \mbox {with } \mu _i \in \mathbb {C} \mbox { distinct}. \end{equation*}

We put $\partial _i:=\dfrac{\partial }{\partial x_i} \,\cdot$ We have an exact sequence

\begin{equation*}0\rightarrow T_X\rightarrow T_{\mathbb {P}|X}\xrightarrow {\ (dq_1,dq_2)\ }\mathscr {O}_X(2)^2\rightarrow 0\,,\end{equation*}

where $dq_i$ maps the restriction of a vector field $V$ on $\mathbb{P}$ to $V\cdot q_i$ . This gives the exact sequence of symmetric tensors

(4) \begin{equation} 0\rightarrow \mathsf{S}^2T_X\rightarrow \mathsf{S}^2 T_{\mathbb{P}|X}\xrightarrow{\ (dq_1,dq_2)\ }T_{\mathbb{P}|X}(2)^2\,, \end{equation}

where $dq_i(V_1V_2)=(V_1\cdot q_i)V_2+(V_2\cdot q_i)V_1$ for $V_1,V_2$ in $H^0(X, T_{\mathbb{P}|X})$ .

Proposition 7.4. The quadratic vector fields $\displaystyle s_i:=\sum _{j\neq i} \dfrac{(x_i\partial _j-x_j\partial _i)^2}{\mu _j-\mu _i}$ in $H^0(X, \mathsf{S}^2 T_{\mathbb{P}|X})$ belong to the image of $H^0(X, \mathsf{S}^2 T_{X})$ .

Proof. According to the exact sequence (4), we have to prove $dq_1(s_i)=dq_2(s_i)=0$ .

We have $(x_i\partial _j-x_j\partial _i)\cdot q_1=0$ ; hence, $dq_1(s_i)=0$ and $dq_2(x_i\partial _j-x_j\partial _i,x_i\partial _j-x_j\partial _i)= 4(\mu _j-\mu _i)x_ix_j(x_i\partial _j-x_j\partial _i)$ . Hence, using $\sum x_j\partial _j=0$ and $q_{1|X}=0$ ,

\begin{equation*} dq_2(s_i)=4x_i^2\sum _{j\neq i} x_j\partial _j-4x_i(\sum _{j\neq i}x_j^2)\partial _i=0\,, \quad \mbox {which proves the proposition.}\end{equation*}

In the rest of this article, we will consider the $s_i$ to be elements of $H^0(X, \mathsf{S}^2 T_{X})$ .

7.3 The double cover

Let $p_0: \mathbb{P}^{n+2}\dashrightarrow \mathbb{P}^{n+1}$ be the projection $(x_0,\ldots, x_{n+2})\mapsto (x_1,\ldots, x_{n+2})$ . The image $p_0(X)$ is the smooth quadric $Q$ in $\mathbb{P}^{n+1}$ defined by

\begin{equation*}\sum _{i=1}^{n+2}(\mu _i-\mu _0)x_{i}^{2}=0\,.\end{equation*}

The restriction $\pi : X\rightarrow Q$ of $p_0$ is a double covering that is branched along the subvariety $B\subset Q$ defined by

\begin{equation*}\sum _{i=1}^{n+2} x_i^2=\sum _{i=1}^{n+2}\mu _i x_i^2=0\,.\end{equation*}

It is a smooth complete intersection of 2 quadrics in $\mathbb{P}^{n+1}$ . The ramification locus $R\subset X$ of $\pi$ (isomorphic to $B$ ) is the hyperplane section $x_0=0$ of $X$ .

The tangent map of $\pi :X\rightarrow Q$ gives a morphism,

\begin{equation*} \tau : T_X\rightarrow \pi ^*T_Q, \end{equation*}

which is an isomorphism outside of $R$ . Consider the normal exact sequence

\begin{equation*}0\rightarrow T_R \rightarrow T_{X|R}\rightarrow N_{R/X}\rightarrow 0\,.\end{equation*}

The involution $\iota :(x_0,\ldots, x_{n+2})\mapsto (-x_0,x_1,\ldots, x_{n+2})$ acts on $T_{X|R}$ ; this splits the exact sequence, giving a decomposition

\begin{equation*}T_{X|R}=T_R\oplus N_{R/X}\end{equation*}

into eigenspaces for the eigenvalues $+1$ and $-1$ . Let $\rho : T_{X|R}\rightarrow T_R$ be the projection on the first summand. We deduce from $\rho$ a sequence of homomorphisms

\begin{equation*}h^k: H^0(X, \mathsf {S}^kT_X)\longrightarrow H^0(X, \mathsf {S}^kT_{X|R})\xrightarrow {\ \mathsf {S}^k\rho \ } H^0(R, \mathsf {S}^kT_R)\,.\end{equation*}

Since $\iota _*\partial _0=-\partial _0$ and $\iota _*\partial _j=\partial _j$ for $j\gt 0$ , we have

(5) \begin{equation} h^2(s_0)=0\quad \mbox{and}\quad h^2(s_i)=\sum _{j\gt 0 \atop j\neq i} \dfrac{(x_i\partial _j-x_j\partial _i)^2}{\mu _j-\mu _i}\quad \mbox{for }i\gt 0\, \end{equation}

in other words, $h^2$ maps $s_1,\ldots, s_{n+2}$ to the elements $\hat{s}_1,\ldots, \hat{s}_{n+2}$ of $H^0(R, \mathsf{S}^2T_R)$ constructed in Proposition 7.2.1 applied to $R$ .

Let $\pi ^*\mathbb{P} T^*Q$ be the pull-back under $\pi$ of the projective bundle $\mathbb{P} T^*Q\rightarrow Q$ . The homomorphism $\tau :T_X\rightarrow \pi ^*T_Q$ gives rise to the birational map $g: \pi ^*\mathbb{P} T^*Q\dashrightarrow \mathbb{P} T^*X$ . Following the geometric description of the tangent map as an elementary transformation of vector bundles in the sense of Maruyama in [Mar72] and [Mar73, Corollary 1.1.1], one has a commutative diagram

(6)

where $p$ and $q$ are the canonical projections; $\nu : \Gamma \rightarrow \mathbb{P} T^* X$ is the blow-up along the subspace $\mathbb{P} T^* R \subset \mathbb{P} T^* X$ defined by the projection $\rho$ ; and $\mu :\Gamma \rightarrow \pi ^*\mathbb{P} T^*Q$ is the blow-up of the image $B'$ of the embedding $B\hookrightarrow \pi ^*\mathbb{P} T^*Q$ deduced from the surjective homomorphism $\pi ^*T_Q\rightarrow \pi ^*N_{B/X}$ .

Let $E_{\mu }$ be the exceptional divisor of $\mu$ . By [Mar73, Theorem 1.1], there is an isomorphism

(7) \begin{equation} \mu ^*\mathscr{O}_{\pi ^*\mathbb{P} T^*Q}(1) \otimes \mathscr{O}_{\Gamma }(-E_{\mu }) \cong \nu ^*\mathscr{O}_{\mathbb{P} T^* X}(1)\, \end{equation}

as well as the equality

(8) \begin{equation} \nu _* E_{\mu } = q^*R\, . \end{equation}

7.4 The divisor of $\boldsymbol{{s}}_{\textbf{0}}$

We now consider the divisor $\mathscr{C}\subset \mathbb{P} T^*Q$ defined in (7.1) and the cartesian diagram

Set $\mathscr{C}^{\,\prime}:= \pi '^{-1}(\mathscr{C})$ . The projection $\mathscr{C}^{\,\prime}\rightarrow X$ is again a smooth quadric fibration, so $\mathscr{C}^{\,\prime}$ is smooth and connected for $n\geq 3$ .

Recall that we have defined the element $\displaystyle s_0:=\sum _{j=1}^{n+2} \dfrac{(x_0\partial _j-x_j\partial _0)^2}{\mu _j-\mu _0} \in H^0(X,\mathsf{S}^2T_X)$ (7.2). We will now view $s_0$ as an element of $H^0(\mathbb{P} T^*X,\mathscr{O}(2))$ .

Proposition 7.5 Assume $n\geq 3$ . We have $g_*\mathscr{C}^{\,\prime}=\mathrm{div}(s_0)$ .

Proof. We first show that $g_*\mathscr{C}^{\,\prime}\in \lvert{\mathscr{O}_{\mathbb{P} T^*X}(2)}\rvert$ . By Proposition 7.1 we have $\mathscr{C}^{\,\prime}\in \lvert{\mathscr{O}_{\pi ^*\mathbb{P} T^*Q}(2)\otimes p^*\mathscr{O}_X(-2)}\rvert$ . Using (7), (8) and the projection formula, we get the linear equivalences

\begin{equation*}\nu _*\mu ^*\mathscr {C}' \sim 2\nu _*\mu ^*(c_1(\mathscr {O}_{\pi ^*\mathbb {P} T^*Q}(1)-p^*R))\sim 2(c_1(\mathscr {O}_{\mathbb {P} T^*X}(1)) +q^*R)-2q^*R=c_1(\mathscr {O}_{\mathbb {P} T^*X}(2))\,.\end{equation*}

Thus, it is enough to prove that $\nu _*\mu ^*\mathscr{C}^{\,\prime}$ is irreducible. Since $\mathscr{C}^{\,\prime}$ is irreducible and $\mu$ is the blow-up along $B'\subset \pi ^*\mathbb{P} T^*Q$ , it suffices to show that $B'$ is not contained in $\mathscr{C}^{\,\prime}$ . If this is the case, then we have $\pi '(B')\subset \pi ' (\mathscr{C}^{\,\prime})=\mathscr{C}$ . But $\pi '(B')=s(B)$ , where $s: B\hookrightarrow \mathbb{P} T^*Q$ is the embedding defined by the surjective homomorphism $T_Q\rightarrow N_{B/Q}$ . Then the result follows from Lemma 7.3.

Since $g_*\mathscr{C}^{\,\prime}$ and $\mathrm{div}(s_0)$ are linearly equivalent effective divisors and $g_*\mathscr{C}^{\,\prime}$ is irreducible, it suffices to show that their restrictions to $\mathbb{P} T_x^*(X)$ coincide at a general point $x\in X$ .

Fix a point $x=(x_0,\dots, x_{n+2})\in X\smallsetminus R$ so that $x_0\not =0$ . Then the tangent map $T\pi (x): T_x(X)\rightarrow T_{\pi (x)}(Q)$ is an isomorphism; in diagram (6), the maps $\mu, \nu$ and $g$ restricted over the fibres at $x$ are all isomorphisms. Let us show that $\mathscr{C}^{\,\prime}$ and $T\pi (\mathrm{div}(s_0))$ define the same quadric in $\mathbb{P} (T_{\pi (x)}(Q))$ .

Note that $\mathscr{C}^{\,\prime}\cap \mathbb{P} (T^*_{x}(X))=\mathscr{C}\cap \mathbb{P} (T^*_{\pi (x)}(Q))$ is the quadric defined by the element $h'_q$ of (7.1). In the coordinates $(z_i)$ defined by $z_i= (\mu _i-\mu _0)^{1/2}x_i$ , the equation of $Q$ is $\displaystyle \sum _{j=1}^{n+2}z_j^2=0$ , so

\begin{equation*}h'_q=\sum _{j=1}^{n+2}\bigl (\dfrac {\partial }{\partial z_j}\bigr )^2= \sum _{j=1}^{n+2}\dfrac {\partial _j^2}{\mu _j-\mu _0}\ \cdot \end{equation*}

On the other hand, since $\pi (x_0,\ldots, x_{n+2})=(x_1,\ldots, x_{n+2})$ , we have $T\pi (\partial _0)=0$ and $T\pi (\partial _j)=\partial _j$ for $j\gt 0$ ; hence,

\begin{equation*}T\pi (s_0)=x_0^2\, \sum _{j=1}^{n+2}\dfrac {\partial _j^2}{\mu _j-\mu _0}\ \cdot \end{equation*}

Since $x_0\neq 0$ , this proves the proposition.

7.5 Proof of part (a) of the theorem

Suppose now that $n \geq 3$ . Consider the double cover $ \pi : X \rightarrow Q$ and the ramification divisor $R \subset X$ . The restriction maps $h^k$ defined in Section 7.3 yield a homomorphism of graded $\mathbb{C}$ -algebras

\begin{equation*} h: S( X) := H^0(X, \mathsf {S}^{\scriptscriptstyle \bullet } T_X) \longrightarrow H^0(R, \mathsf {S}^{\scriptscriptstyle \bullet } T_{R}) =: S( R). \end{equation*}

Proposition 7.6 The kernel $\mathscr{I}$ of $h$ is the ideal generated by $s_0$ .

Proof. Since $\mathscr{I}$ is a homogeneous ideal, it suffices to prove that every homogeneous element $s\in \mathscr{I}$ can be written as $s=s' s_0$ for some element $s'\in S(X)$ .

Choose an element $s \in \mathscr{I}$ of degree $k$ . This element corresponds to an effective Cartier divisor $G$ in the linear system $|\mathscr{O}_{\mathbb{P} T^* X}(k)|$ . Recall the commutative diagram (6):

Choose $ \hat{G} :=\mu _* \nu ^* G\subset \pi ^*\mathbb{P} T^*Q$ . By (7), $\hat{G}$ belongs to the linear system $\lvert{\mathscr{O}_{\pi ^*\mathbb{P} T^*Q}(k)}\rvert$ .

Here is the key observation: Since $s \in \mathscr{I}$ , the divisor $ \hat{G} \subset \pi ^*\mathbb{P} T^*Q$ contains $p^* R$ . Indeed, since $(\pi ^*T_Q)_{|R}$ is invariant under $\iota$ , the homomorphism $\tau ^{} _{|R}$ factors as

\begin{equation*}\tau ^{} _{|R}:T_{X|R}\xrightarrow {\ \rho \ }T_R \longrightarrow (\pi ^*T_Q)_{|R}\,.\end{equation*}

Therefore, we have a commutative diagram,

and $\mathsf{S}^k\tau (s)$ vanishes on $R$ . But $\hat{G}$ is the divisor of $\mathsf{S}^k\tau (s)$ , viewed as a section of $\mathscr{O}_{\pi ^*\mathbb{P} T^*Q}(k)$ ; hence, $\hat{G}$ contains $p^*R$ .

Now we want to show that the divisor $\mathscr{C}^{\,\prime}\subset \pi ^*\mathbb{P} T^*Q$ is a component of $\hat{G}-p^*R$ . Recall from (7.1) that $\mathscr{C}$ is the union of the lines $\ell$ that are contracted by the morphism $\gamma : \mathbb{P} T^*Q\rightarrow \mathbb{G}$ and that $c_1(\mathscr{O}_{\mathbb{P} T^*Q}(1))\cdot \ell =0$ . Thus the curves $\ell ':=\pi '^*\ell$ cover $\mathscr{C}^{\,\prime}$ and satisfy $c_1(\mathscr{O}_{\pi ^*\mathbb{P} T^*Q}(1))\cdot \ell '=0$ . On the other hand, the divisor $R\subset X$ is a hyperplane section, so $p^*R\cdot \ell '=R\cdot p_*\ell '\gt 0$ . Therefore,

\begin{equation*}(\hat {G}-p^*R)\cdot \ell '\lt 0\,, \end{equation*}

so $\mathscr{C}^{\,\prime}$ is a component of $\hat{G}$ . Thus, $g_*\mathscr{C}^{\,\prime}$ is a component of $G$ . Since $g_*\mathscr{C}^{\,\prime}=\mathrm{div}(s_0)$ by Proposition 7.5, this proves the proposition.

The following proposition implies part (a) of our main theorem:

Proposition 7.7. Assume $n\geq 2$ . For any choice of indices $0\leq i_1\lt \ldots \lt i_n\leq n+2$ , the homomorphism $\mathbb{C}[t_1,\ldots, t_{n}]\rightarrow S(X)$ , which maps $t_j$ to $s_{i_{j}}$ , with $\deg (t_i)=2$ , is an isomorphism of graded $\mathbb{C}$ -algebras.

Proof. We argue by induction on $n$ . The statement for $n=2$ follows from [DOL19, Theorem 5.1], except for the fact that any two of the $s_i$ generate $H^0(X,\mathsf{S}^2T_X)$ . Up to the permuting of the coordinates, it suffices to prove that $s_0$ and $s_1$ are linearly independent. But $h^2:H^0(X,\mathsf{S}^2T_X)\rightarrow H^0(R,\mathsf{S}^2T_R)$ maps $s_0$ to zero and maps $s_i$ (for $i\gt 0$ ) to the corresponding elements $\hat{s}_i$ of $H^0(R,\mathsf{S}^2T_R)$ ; this implies our assertion.

Assume $n\geq 3$ . By the induction hypothesis, the homomorphism $\mathbb{C}[t_1,\ldots, t_{n-1}]\rightarrow S(R)$ , which maps $t_i$ to $\hat{s_i}$ , is an isomorphism of graded $\mathbb{C}$ -algebras (with $\deg (t_i)=2$ ). It follows that $h$ is surjective and that $(s_0,\ldots, s_{n-1})$ form a basis of $H^0(X,\mathsf{S}^2 T_X)$ and generate the $\mathbb{C}$ -algebra $S(X)$ . Thus we have a surjective homomorphism $u:\mathbb{C}[t_0,\ldots, t_{n-1}]\rightarrow S(X)$ , with $u(t_i)=s_i$ .

In particular, the Krull dimension of $S(X)$ is at most $n$ . On the other hand, the ring $S(X)$ is a domain, and $s_0$ is neither zero nor a unit. Thus, by Krull’s Hauptidealsatz, the Krull dimension of $S(X)$ is equal to $n$ ; hence, $u$ is an isomorphism. By permutation of the coordinates, we get the same result for any choice of $n$ elements in $\{s_0,\ldots, s_{n+2}\}$ . This proves the proposition.