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 [Reference De Oliveira and LangdonDOL19], while (b) and (c) are proved in [Reference Kim and LeeKL22]. 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
$\mathbb{P}^1$
[Reference CasagrandeCas15], so
$\Phi :T^*X\rightarrow \mathbb{C}^2$
is identified to the Hitchin fibration (see [Reference Biswas, Holla and KumarBHK10]).
For
$n=3$
,
$X$
is isomorphic to the moduli space of vector bundles of rank 2 and fixed determinant of odd degree [Reference NewsteadNew68]; 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, [Reference AudinA96].
A classical example, the geodesic flow on an ellipsoid, is discussed in detail in [Reference KnörrerK80]. 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 [Reference KnörrerK80] (with a different purpose).
(3) Such a situation is rather exceptional: Most varieties do not admit nonzero symmetric tensors (for instance, hypersurfaces of degree
$\geq 3$
[Reference Liu and ShaoHLS22]); when they do, even for varieties as simple as quadrics, the algebra of symmetric tensors is fairly complicated (see, for instance, [Reference Beauville and LiuBLi24]). 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
$N$
[Reference NewsteadNew68]. 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 [Reference HitchinHit87].
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, [Reference Beauville, Narasimhan and RamananBNR89]). 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
$f$
[Reference Griffiths and HarrisGH79]: 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 [Reference ReidReid72, 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 ([Reference ReidReid72]; see also [Reference Desale and RamananDR76]). 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 (-1JC). 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
$T_2\times \{\pm 1_{JC}\}$
[Reference Desale and RamananDR76, 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
\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. [Reference Birkenhake and LangeBL92, 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
\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}
\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 [Reference Liu and ShaoHLS22]. 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 [Reference Anella and HöringAH23]).
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
\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
\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 [Reference MaruyamaMar72] and [Reference MaruyamaMar73, Corollary 1.1.1], one has a commutative diagram
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 [Reference MaruyamaMar73, Theorem 1.1], there is an isomorphism
\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
\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 [Reference De Oliveira and LangdonDOL19, 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.
Conflicts of Interest
None.
Financial Support
J. Liu is supported by the National Key Research and Development Program of China (No. 2021YFA1002300), the NSFC grants (No. 12001521 and No. 12288201) and the CAS Project for Young Scientists in Basic Research (No. YSBR-033).
Journal Information
Moduli is published as a joint venture of the Foundation Compositio Mathematica and the London Mathematical Society. As not-for-profit organisations, the Foundation and Society reinvest
$100\%$
of any surplus generated from their publications back into mathematics through their charitable activities.
Open access
