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The Hodge diamond of O’Grady’s six-dimensional example

Published online by Cambridge University Press:  21 March 2018

Giovanni Mongardi
Affiliation:
Dipartimento di Matematica, Alma Mater Studiorum Università di Bologna, Piazza di Porta San Donato 5, Bologna 40126, Italia email giovanni.mongardi2@unibo.it
Antonio Rapagnetta
Affiliation:
Dipartimento di Matematica, Università di Roma Tor Vergata, Via della Ricerca Scientifica 1, Roma 00133, Italia email rapagnet@axp.mat.uniroma2.it
Giulia Saccà
Affiliation:
Department of Mathematics, Stony Brook University, Stony Brook, NY 11974-3651, USA email giulia.sacca@stonybrook.edu
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Abstract

We realize O’Grady’s six-dimensional example of an irreducible holomorphic symplectic (IHS) manifold as a quotient of an IHS manifold of $\text{K3}^{[3]}$ type by a birational involution, thereby computing its Hodge numbers.

Type
Research Article
Copyright
© The Authors 2018 

1 Introduction

In this paper we present a new way of obtaining O’Grady’s six-dimensional example of an irreducible holomorphic symplectic manifold and use this to compute its Hodge numbers. Further applications, such as the description of the movable cone or the answer to Torelli-type questions for this deformation class of irreducible holomorphic symplectic manifolds, will be the topic of a subsequent paper.

Recall that irreducible holomorphic symplectic (IHS) manifolds are simply connected compact Kähler manifolds that have a holomorphic symplectic form, unique up to scalars. They arise naturally as one of the three building blocks of manifolds with trivial first Chern class according to the Beauville–Bogomolov decomposition [Reference BogomolovBog78, Reference BeauvilleBea83], the other two blocks being abelian varieties and Calabi–Yau manifolds. By definition, IHS manifolds are higher-dimensional generalizations of K3 surfaces; moreover, they have a canonically defined quadratic form on their integral second cohomology group, which allows one to speak of their periods and to develop their theory in a way which is analogous to the theory of K3 surfaces. The interested reader can see [Reference HuybrechtsHuy99] and [Reference O’GradyO’Gr12] for a general introduction on the topic.

Two deformation classes of IHS manifolds in every even dimension greater than two were introduced by Beauville in [Reference BeauvilleBea83]. They are the Hilbert scheme of $n$ points on a K3 and the generalized Kummer variety of dimension $2n$ of an abelian surface (i.e. the Albanese fiber of the Hilbert scheme of $n+1$ points of the abelian surface). Elements of these two deformation classes have second Betti number equal to $23$ and $7$ , respectively, and are referred to as IHS manifolds of K3 $^{[n]}$ type and of generalized Kummer type, respectively. There are two more examples, found by O’Grady in [Reference O’GradyO’Gr99] and [Reference O’GradyO’Gr03], of dimensions ten and six, respectively, which are obtained from a symplectic resolution of certain singular moduli spaces of sheaves on a K3 surface and on an abelian surface, respectively. They are referred to as the exceptional examples of IHS, and their deformation classes are denoted by OG10, respectively OG6.

These exceptional examples have not been studied as much as IHSs of K3 $^{[n]}$ type and of generalized Kummer type and their geometry is less well understood. Though their topological Euler characteristic is known (see [Reference RapagnettaRap04] and [Reference MozgovyyMoz06]), even other basic invariants such as their Hodge numbers have not been computed yet. In the case of manifolds of K3 $^{[n]}$ type or of generalized Kummer type, the Hodge numbers were computed by Göttsche in [Reference GöttscheGot90] and Göttsche and Sörgel in [Reference Göttsche and SörgelGS93].

One of the main results of this paper is to realize O’Grady’s six-dimensional example as a quotient of an IHS manifold of K3 $^{[3]}$ type by a birational symplectic involution: we therefore relate this deformation class to the most studied deformation class of IHS manifolds and this allows us, by resolving the indeterminacy locus of the involution and by describing explicitly its fixed locus (which has codimension 2), to compute the Hodge numbers. The involution we use was first introduced in [Reference RapagnettaRap04] in order to compute the Beauville–Bogomolov form for IHS of type OG6 and then used in [Reference Mongardi and WandelMW17] to determine a special subgroup of the automorphisms group of such manifolds. The main result of the present paper is the following.

Theorem 1.1. Let $\widetilde{K}$ be an irreducible holomorphic symplectic of type OG6. The odd Betti numbers of $\widetilde{K}$ are zero, and its non-zero Hodge numbers are collected in the following table:

$$\begin{eqnarray}\begin{array}{@{}ccccccl@{}} & & & H^{0,0}=1 & & & \\ & & H^{2,0}=1 & H^{1,1}=6 & H^{0,2}=1 & & \\ & H^{4,0}=1 & H^{3,1}=12 & H^{2,2}=173 & H^{1,3}=12 & H^{0,4}=1 & \\ H^{6,0}=1 & H^{5,1}=6 & H^{4,2}=173 & H^{3,3}=1144 & H^{2,4}=173 & H^{1,5}=6 & H^{0,6}=1\\ & H^{6,2}=1 & H^{5,3}=12 & H^{4,4}=173 & H^{3,5}=12 & H^{2,6}=1 & \\ & & H^{6,4}=1 & H^{5,5}=6 & H^{4,6}=1 & & \\ & & & H^{6,6}=1. & & & \end{array}\end{eqnarray}$$

As a corollary, we also get the Chern numbers of this sixfold, see Proposition 6.8 for details.

Let us outline the main ideas that go into the proof of our main result.

Recall that O’Grady’s six-dimensional example is obtained as a symplectic resolution of a certain natural subvariety of a moduli space of sheaves on an abelian surface $A$ . In order to describe how to obtain it as a ‘quotient’ of another IHS by a birational symplectic automorphism, we first need to introduce some notation.

Let $X$ be a K3 or an abelian surface. Fix an effective Mukai vectorFootnote 1   $v\in H_{\text{alg}}^{\ast }(X,\mathbb{Z})$ , with $v^{2}\geqslant -2$ , and let $H$ be a sufficiently general ample line bundle on $X$ . It is well known [Reference MukaiMuk88, Reference YoshiokaYos07] that, if $v$ is primitive, the moduli space $M_{v}(X,H)$ of $H$ -stable sheaves on $X$ with Mukai vector $v$ is a smooth projective manifold of dimension $v^{2}+2$ and that, if $v^{2}\geqslant 0$ , it admits a holomorphic symplectic form. If $X$ is a K3 surface, then $M_{v}(X,H)$ is an IHS variety of K3 $^{[n]}$ type, for $n=v^{2}/2+1$ . Whereas, if $X=A$ is an abelian surface and if $v^{2}\geqslant 4$ , there is a nontrivial Albanese variety and, in order to get an irreducible holomorphic symplectic manifold, one needs to consider a fiber

(1.1) $$\begin{eqnarray}K_{v}(A,H):=\operatorname{alb}^{-1}(0)\end{eqnarray}$$

of the Albanese morphism (which is isotrivial)

$$\begin{eqnarray}\operatorname{alb}:M_{v}(A,H)\rightarrow A\times A^{\vee }.\end{eqnarray}$$

Recall that if $v^{2}\geqslant 6$ , $K_{v}(A,H)$ is deformation equivalent to the generalized Kummer variety $K^{[n]}(A):=\sum ^{-1}(0)$ , where $\sum :A^{[n+1]}\rightarrow A$ is the summation morphism.

If we consider an $H$ -stable sheaf $F$ with a primitive Mukai vector $v_{0}$ , then for $m\geqslant 2$ , the sheaf $F^{\oplus m}$ is strictly $H$ -semistable. If we set $v=mv_{0}$ and assume $v_{0}^{2}>0$ , this sheaf determines a singular point of the moduli space $M_{v}(X,H)$ , whose smooth locus still carries a holomorphic symplectic form. In [Reference O’GradyO’Gr99] and [Reference O’GradyO’Gr03], O’Grady considered the case of $v_{0}=(1,0,-1)$ and $m=2$ , and showed that the singular symplectic variety $M_{v}(X,H)$ admits a symplectic resolution $\widetilde{M}_{v}(X,H)$ . For $X$ K3, this resolution gives a 10-dimensional IHS manifold of type OG10. For $X=A$ , fix $F_{0}\in M_{v}(A,H)$ and denote by $K_{v}(A,H)$ the fiber over 0 of the isotrivial fibration

(1.2) $$\begin{eqnarray}\begin{array}{@{}cccc@{}}\mathbf{a}_{v}: & M_{v}(A,H) & \longrightarrow & A\times A^{\vee },\\ & F & \longmapsto & (\text{Alb}(c_{2}(F)),\det (F)\otimes \det (F_{0})^{-1}),\end{array}\end{eqnarray}$$

where $\text{Alb}:CH_{0}(A)\rightarrow A$ is the Albanese homomorphism. The proper transform $\widetilde{K}_{v}(A,H)$ of $K_{v}(A,H)$ in $\widetilde{M}_{v}(X,H)$ is smooth and the induced map

(1.3) $$\begin{eqnarray}f_{v}:\widetilde{K}_{v}(A,H)\rightarrow K_{v}(A,H)\end{eqnarray}$$

is a symplectic resolution. The gives the six-dimensional IHS $\widetilde{K}_{v}(A,H)$ , whose deformation type is called OG6, that is the object of this paper.

Lehn and Sorger proved in [Reference Lehn and SorgerLS06] that for any primitive $v_{0}$ , with $v_{0}^{2}=2$ , the moduli space $M_{2v_{0}}(X,H)$ admits a symplectic resolution. Finally, Perego and the second named author [Reference Perego and RapagnettaPR13] showed that for any choice of $v_{0}$ , with $v_{0}^{2}=2$ , on a K3 or abelian surface, the IHS manifolds that one gets are deformation equivalent to OG10 and OG6, respectively.

When $A$ is a general principally polarized abelian surface and $M_{v}(X,H)$ parametrizes pure one-dimensional sheaves, the IHS manifold $\widetilde{K}_{v}(A,H)$ is the image of a degree-2 rational map whose domain is an IHS manifold of K3 $^{[3]}$ type as we now briefly sketch: this is the starting point of our proof.

Let us consider a principal polarization $\unicode[STIX]{x1D6E9}\subset A$ . The Mukai vector $v_{0}=(0,\unicode[STIX]{x1D6E9},1)$ satisfies $v_{0}^{2}=2$ , and hence, if we set $v=2v_{0}$ , there is symplectic resolution $\widetilde{K}_{v}(A,H)\rightarrow K_{v}(A,H)$ that is deformation equivalent to OG6. There is a natural support morphism $K_{v}(A,H)\rightarrow |2\unicode[STIX]{x1D6E9}|=\mathbb{P}^{3}$ , realizing $K_{v}(A,H)$ as a Lagrangian fibration. By definition of $K_{v}(A,H)$ , the fiber over a smooth curve $C\in |2\unicode[STIX]{x1D6E9}|$ is the kernel of the natural morphism $\operatorname{Pic}^{6}(C)\rightarrow A$ (which is also the restriction of $\mathbf{a}_{v}$ to $\operatorname{Pic}^{6}(C)\subset M_{v}(A,H)$ ).

It is well known that the morphism associated to the linear system $|2\unicode[STIX]{x1D6E9}|$ is the quotient morphism $A\rightarrow A/\pm 1\subset \mathbb{P}^{3}$ onto the singular Kummer surface of $A$ . Let $S\rightarrow A/\pm 1$ be the minimal resolution of $A$ . It is well known that $S$ , the Kummer surface of $A$ , is a K3 surface. Note that $S$ comes naturally equipped with the degree-4 nef line bundle $D$ obtained by pulling back the hyperplane section of $A/\pm 1\subset \mathbb{P}^{3}$ . Consider the diagram

(1.4)

where $\widetilde{A}$ is the blow up of $A$ at its 16 2-torsion points or, equivalently, the ramified cover of $S$ along the exceptional curves $E_{1},\ldots ,E_{16}$ of $p$ . Consider the moduli space $M_{w}(S)$ of sheaves on $S$ with Mukai vector $w=(0,D,1)$ that are stable with respect to a chosen, sufficiently general, polarization. This is an IHS manifold birational to the Hilbert cube of $S$ and it has a natural morphism $M_{w}(S)\rightarrow |D|=\mathbb{P}^{3}$ realizing it as the relative compactified Jacobian of the linear system $|D|$ (also a Lagrangian fibration).

The morphisms in diagram (1.4) induce a rational map

$$\begin{eqnarray}\unicode[STIX]{x1D6F7}:M_{w}(S){\dashrightarrow}K_{v}(A,H)\subset M_{v}(A,H),\end{eqnarray}$$

which maps the generic $[F]\in M_{w}(S)$ to $[q^{\ast }p_{\ast }F]\in M_{v}(A,H)$ and is generically $2:1$ onto its image. Since $M_{w}(S)$ is simply connected, the image of this map lies in a fiber of $\mathbf{a}_{v}$ , giving a generically $2:1$ map $\unicode[STIX]{x1D6F7}:M_{w}(S){\dashrightarrow}K_{v}(A,H)$ .

The map $\unicode[STIX]{x1D6F7}$ commutes with the Lagrangian fibrations on $M_{w}(S)$ and $K_{v}(A,H)$ . If $C^{\prime }$ is a smooth curve in $|D|$ , $\operatorname{Pic}^{3}(C^{\prime })$ embeds into $M_{w}(S)$ as a fiber of the Lagrangian fibration. If $C\in |2\unicode[STIX]{x1D6E9}|$ is the étale cover of $C^{\prime }$ , analogously $\ker [\operatorname{Pic}^{6}(C)\rightarrow A]$ embeds into $K_{v}(A,H)$ as a fiber of the Lagrangian fibration. On these fibers, the map $\unicode[STIX]{x1D6F7}$ restricts to the natural $2:1$ pull back morphism $\operatorname{Pic}^{3}(C^{\prime })\rightarrow \operatorname{Pic}^{6}(C)$ , whose image is precisely $\ker [\operatorname{Pic}^{6}(C)\rightarrow A]$ . Recall that $\sum _{i}E_{i}$ is divisible by 2 in $H^{2}(S,\mathbb{Z})$ and that the line bundle $\unicode[STIX]{x1D702}:={\mathcal{O}}_{S}(\frac{1}{2}\sum E_{i})$ determines the double cover  $q$ . It follows that the involution on $M_{w}(S)$ corresponding to $\unicode[STIX]{x1D6F7}$ is given by tensoring by $\unicode[STIX]{x1D702}$ and $\widetilde{K}_{v}(A,H)$ is a birational model of the ‘quotient’ of $M_{w}(S)$ by the birational involution induced by tensorization by $\unicode[STIX]{x1D702}$ .

In this paper, for any abelian surface $A$ and for an effective Mukai vector $v=2v_{0}$ with $v_{0}^{2}=2$ on $A$ , we show that $\widetilde{K}_{v}(A,H)$ admits a rational double cover from an IHS manifold $\text{}\underline{Y}_{v}(A,H)$ of K3 $^{[3]}$ type. Recall that the singular locus $\unicode[STIX]{x1D6F4}_{v}\subset K_{v}(A,H)$ has codimension $2$ and can be identified with $(A\times A^{\vee })/\pm 1$ (for more details, see § 2). Following [Reference O’GradyO’Gr03], the symplectic resolution (1.3) can be obtained by two subsequent blow ups followed by a contraction: first, one blows up the singular locus of $\unicode[STIX]{x1D6F4}_{v}$ , then one blows up the proper transform of $\unicode[STIX]{x1D6F4}_{v}$ itself (which is smooth). These two operations produce a manifold $\widehat{K}_{v}(A,H)$ that has a holomorphic two form degenerating along the strict transform of the exceptional divisor of the first blow up; contracting this exceptional divisor finally gives the manifold $\widetilde{K}_{v}(A,H)$ that has a symplectic two form and a regular morphism $\widetilde{K}_{v}(A,H)\rightarrow K_{v}(A,H)$ , which is, therefore, a symplectic resolution. The inverse image $\widetilde{\unicode[STIX]{x1D6F4}}_{v}$ of $\unicode[STIX]{x1D6F4}_{v}$ in $\widetilde{K}_{v}(A,H)$ is an irreducible divisor, which is divisible by two in the integral cohomology by results of the second named author [Reference RapagnettaRap04]. We show that the associated ramified double cover is a projective variety birational to an IHS manifold of K3 $^{[3]}$ type, which we denote by $\text{}\underline{Y}_{v}(A,H)$ and which is equipped with a birational symplectic involution.

This enables us to reconstruct $\widetilde{K}(A,H)$ starting from $\text{}\underline{Y}_{v}(A,H)$ , and its symplectic birational involution

$$\begin{eqnarray}\text{}\underline{\unicode[STIX]{x1D70F}}_{v}:\text{}\underline{Y}_{v}(A,H){\dashrightarrow}\text{}\underline{Y}_{v}(A,H).\end{eqnarray}$$

More specifically, $\text{}\underline{Y}_{v}(A,H)$ contains 256 $\mathbb{P}^{3}$ , the birational involution $\text{}\underline{\unicode[STIX]{x1D70F}}_{v}$ is regular on the complement of these $\mathbb{P}^{3}$ , and, moreover, this involution lifts to a regular involution on the blow up $\overline{Y}_{v}(A,H)$ of $\text{}\underline{Y}_{v}(A,H)$ along the 256 $\mathbb{P}^{3}$ . The fixed locus $\overline{\unicode[STIX]{x1D6E5}}_{v}$ of the induced involution on $\overline{Y}_{v}(A,H)$ is smooth and four-dimensional, hence the blow up $\widehat{Y}_{v}(A,H)$ of $\overline{Y}_{v}(A,H)$ along this fixed locus carries an involution $\widehat{\unicode[STIX]{x1D70F}}_{v}$ admitting a smooth quotient $\widehat{Y}_{v}(A,H)/\widehat{\unicode[STIX]{x1D70F}}_{v}$ . This quotient is $\widehat{K}_{v}(A,H)$ , and $\widehat{Y}_{v}(A,H)$ is its double cover branched over $\widehat{\unicode[STIX]{x1D6F4}}_{v}$ . Finally, $\widehat{K}_{v}(A,H)$ is the blow up of $\widetilde{K}_{v}(A,H)$ along $256$ smooth three-dimensional quadrics.

This construction allows one to relate the Hodge numbers of $\widetilde{K}_{v}(A,H)$ to the invariant Hodge numbers of $\text{}\underline{Y}_{v}(A,H)$ . Finally, the invariant Hodge numbers of $\text{}\underline{Y}_{v}(A,H)$ may be determined by using monodromy results of Markman [Reference MarkmanMar02]. In particular, the computation of the Betti numbers can be carried out more easily by observing that $\overline{\unicode[STIX]{x1D6E5}}_{v}\subset \overline{Y}_{v}(A,H)$ is a smooth submanifold with vanishing rational odd cohomology. As the odd Betti numbers of $\text{}\underline{Y}_{v}(A,H)$ are zero, the same holds for the odd Betti numbers of $\widehat{Y}_{v}(A,H)$ . Since the rational cohomology of $\widetilde{K}_{v}(A,H)$ injects into the rational cohomology of $\widehat{Y}_{v}(A,H)$ , the odd Betti numbers of $\widetilde{K}_{v}(A,H)$ are zero too. Since the Betti numbers $b_{0}$ and $b_{2}$ are known, it remains to determine $b_{4}$ and $b_{6}$ : this can be done by using the knowledge of the Euler characteristic and Salomon’s universal relation on Betti numbers of IHS manifolds (see Proposition 6.1).

We should point out that this construction cannot be carried out for IHS manifolds of type OG10, since the exceptional divisor of the second blow up (the procedure to obtain the symplectic resolution is the same) is not divisible by 2 in the integral cohomology (see [Reference RapagnettaRap08] for the proof).

Remark 1.2. Recently, there has been considerable interest in exhibiting and classifying symplectic automorphisms of IHS manifolds [Reference Boissiére and SartiBS12, Reference Höhn and MasonHM14, Reference MenetMen15, Reference MongardiMon16]. Note that quotients of IHS by symplectic automorphisms rarely admit a symplectic resolution since for this to happen the fixed locus has to be of codimension $2$ (see [Reference KawataniKaw09] for one of the few cases where this happens). Our construction, however, indicates that ‘quotients’ by birational symplectic automorphisms can have a symplectic resolution, and thus they are potentially interesting. In upcoming work, we will study some of these birational morphisms for manifolds of K3 $^{[n]}$ type.

The structure of the paper is as follows. In § 2, we recall local and global properties of O’Grady’s and Lehn–Sorger symplectic resolution. In § 3, we construct an affine double of the Lehn–Sorger local model of the deepest stratum of the singularity of $K_{v}(A,H)$ , branched over the singular locus. In § 4, we globalize the previous results to construct global double covers $Y_{v}$ of $K_{v}(A,H)$ branched over the singular locus. In § 5, we prove that $Y_{v}$ is birational to an IHS manifold of K3 $^{[3]}$ type. Finally, in § 6, we use the previous results to compute the Hodge numbers.

Notation

For a closed embedding $X_{1}\subset X_{2}$ of algebraic varieties we denote with $Bl_{X_{1}}X_{2}$ the blow up of $X_{2}$ along $X_{1}$ .

For a vector bundle $X_{3}\rightarrow Y$ , we denote by $\mathbb{P}(X_{3})\rightarrow Y$ its projectification.

For an affine cone $X_{4}$ we denote by $\mathbb{P}(X_{4})$ its projectification.

Finally we denote by $H^{k}(X_{1})$ the $k$ th singular cohomology group of $X_{1}$ with rational coefficient and by $h^{k}(X_{1})$ its dimension.

2 The resolution

Let us fix a primitive Mukai vector $v_{0}\in H_{\text{alg}}^{\ast }(A,\mathbb{Z})$ with $v_{0}^{2}=2$ , set $v=2v_{0}$ , and consider a $v$ -generic ample line bundle $H$ on $A$ (see [Reference Perego and RapagnettaPR13, § 2.1]). By [Reference Lehn and SorgerLS06, Théorème 1.1] the projective variety $K_{v}:=K_{v}(A,H)$ admits a symplectic resolution $\widetilde{K}_{v}$ which is deformation equivalent to O’Grady’s six-dimensional example by [Reference Perego and RapagnettaPR13, Theorem 1.6(2)]. In this section we recall the description of the singularities of $K_{v}$ and of the symplectic resolution $f:\widetilde{K}_{v}\rightarrow K_{v}$ following both the papers of O’Grady [Reference O’GradyO’Gr99, Reference O’GradyO’Gr99] and Lehn and Sorger [Reference Lehn and SorgerLS06].

Since the singular locus $\unicode[STIX]{x1D6F4}_{v}$ of $K_{v}$ parametrizes polystable sheaves of the form $F_{1}\oplus F_{2}$ , with $F_{i}\in M_{v_{0}}(X,H)$ , we have $\unicode[STIX]{x1D6F4}_{v}=K_{v}\cap \operatorname{Sym}^{2}M_{v_{0}}(A,H)$ . Since $v_{0}^{2}=2$ the smooth moduli space $M_{v_{0}}$ is isomorphic to $A\times A^{\vee }$ and, as the Albanese map $\operatorname{alb}$ is an isotrivial fibration, the singular locus $\unicode[STIX]{x1D6F4}_{v}$ is isomorphic to $(A\times A^{\vee })/\pm 1$ . This also implies that the singular locus $\unicode[STIX]{x1D6FA}_{v}$ of $\unicode[STIX]{x1D6F4}_{v}$ consists of 256 points representing sheaves of the form $F^{\oplus 2}$ with $F\in M_{v_{0}}(X,H)$ .

The analytic type of the singularities appearing in $K_{v}$ is completely understood. If $p\in \unicode[STIX]{x1D6F4}_{v}\setminus \unicode[STIX]{x1D6FA}_{v}$ , i.e. $p$ represents a polystable sheaf of the form $F_{1}\oplus F_{2}$ where $F_{1}\neq F_{2}$ , there exists a neighborhood $U\subset K_{v}$ of $p$ , in the classical topology, biholomorphic to a neighborhood of the origin in the hypersurface defined in $\mathbb{A}^{7}$ by the equation $\sum _{i=1}^{3}x_{i}^{2}=0$ (see for example [Reference Arbarello and SaccàAS18, Proposition 4.4] or [Reference O’GradyO’Gr99, Proposition 1.4.1]), i.e. $K_{v}$ has an $A_{1}$ singularity along $\unicode[STIX]{x1D6F4}_{v}\setminus \unicode[STIX]{x1D6FA}_{v}$ .

If $p\in \unicode[STIX]{x1D6FA}_{v}$ , the description of the analytic type of the singularity of $K_{v}$ at $p$ is due to Lehn and Sorger and it is contained in [Reference Lehn and SorgerLS06, Théorème 4.5]. To recall this description, let $V$ be a four-dimensional vector space, let $\unicode[STIX]{x1D70E}$ be a symplectic form on $V$ , and let $\mathfrak{sp}(V)$ be the symplectic Lie algebra of $(V,\unicode[STIX]{x1D70E})$ , i.e. the Lie algebra of the Lie group of the automorphisms of $V$ preserving the symplectic form $\unicode[STIX]{x1D70E}$ .

We let

$$\begin{eqnarray}Z:=\{A\in \mathfrak{sp}(V)\,|\,A^{2}=0\}\end{eqnarray}$$

be the subvariety of matrices in $\mathfrak{sp}(V)$ having square zero. It is known that $Z$ is the closure of the nilpotent orbit of type $\mathfrak{o}(2,2)$ , which parametrizes rank- $2$ square-zero matrices. Moreover, by Criterion $2$ of [Reference HesselinkHes79], $Z$ is also a normal variety.

By [Reference Lehn and SorgerLS06, Théorèm 4.5], if $p\in \unicode[STIX]{x1D6FA}_{v}$ , there exists a Euclidean neighborhood of $p$ in $K_{v}$ , biholomorphic to a neighborhood of the origin in $Z$ . Hence the local geometry of a symplectic desingularization of $K_{v}$ is encoded in the local geometry of a symplectic desingularization of  $Z$ .

Let $\unicode[STIX]{x1D6F4}$ be the singular locus of $Z$ and let $\unicode[STIX]{x1D6FA}$ be the singular locus of $\unicode[STIX]{x1D6F4}$ . Let us recall that $\dim Z=6$ , $\dim \unicode[STIX]{x1D6F4}=4$ , $\dim \unicode[STIX]{x1D6FA}=0$ and, more precisely,

$$\begin{eqnarray}\unicode[STIX]{x1D6F4}=\{A\in Z\,|\operatorname{rk}A\leqslant 1\}\quad \text{and}\quad \unicode[STIX]{x1D6FA}=\{0\}.\end{eqnarray}$$

Let $G\subset \text{Gr}(2,V)\subset \mathbb{P}(\wedge ^{2}V)$ be the Grassmannian of Lagrangian subspaces of $V$ , note that $G$ is a smooth three-dimensional quadric and set

$$\begin{eqnarray}\widetilde{Z}:=\{(A,U)\,|\,A(U)=0\}\subset Z\times G.\end{eqnarray}$$

The restriction $\unicode[STIX]{x1D70B}_{G}:\widetilde{Z}\rightarrow G$ of the second projection of $Z\times G$ makes $\widetilde{Z}$ the total space of a three-dimensional vector bundle, the cotangent bundle of $G$ . In particular, $\widetilde{Z}$ is smooth and the restriction

$$\begin{eqnarray}f:\widetilde{Z}\rightarrow Z\end{eqnarray}$$

of the first projection of $Z\times G$ , which is an isomorphism when restricted to the locus of rank- $2$ matrices, is a resolution of the singularities. The fiber $f^{-1}(A)$ , over a point $A\in \unicode[STIX]{x1D6F4}$ , is the $\mathbb{P}^{1}$ parametrizing Lagrangian subspaces contained in the three-dimensional kernel of $A$ and the central fiber $f^{-1}(0)$ is the whole $G$ . As $Z$ has an $A_{1}$ singularity along $\unicode[STIX]{x1D6F4}\setminus \unicode[STIX]{x1D6FA}$ and $G$ has dimension $3$ , it follows that $f:\widetilde{Z}\rightarrow Z$ is a symplectic resolution.

Remark 2.1. Let ${\mathcal{U}}\subset V\otimes {\mathcal{O}}_{G}$ be the rank- $2$ tautological bundle. The smooth symplectic variety $\widetilde{Z}$ is isomorphic to the total space $\mathbf{Sym}_{G}^{2}\,{\mathcal{U}}$ of the second symmetric power $\operatorname{Sym}_{G}^{2}{\mathcal{U}}$ of ${\mathcal{U}}$ . In fact, an endomorphism $A\in \mathfrak{gl}(V)$ belongs to $Z$ if and only if the following conditions hold:

  1. (1) $A^{2}=0$ ;

  2. (2) $\unicode[STIX]{x1D70E}(Av_{1},v_{2})=\unicode[STIX]{x1D70E}(Av_{2},v_{1})$ for any $v_{1},v_{2}\in V$ .

By $(2)$ the kernel $\ker A$ and the image $\operatorname{Im}A$ of $A$ are orthogonal with respect to $\unicode[STIX]{x1D70E}$ . Hence, for $(A,U)\in \widetilde{Z}$ , we have $V\rightarrow \operatorname{Im}A\subset U\subset \ker A\subset V$ . Since $U\subset V$ is Lagrangian we have $V/U\cong U^{\vee }$ , so $A$ has a factorization of the form $V{\twoheadrightarrow}U^{\vee }\rightarrow U{\hookrightarrow}V$ . Moreover the induced linear map $\unicode[STIX]{x1D711}_{A}\in \operatorname{Hom}(U^{\vee },U)=U\otimes U$ defines a bilinear form on $U^{\vee }$ that is symmetric if and only if $(2)$ holds.

Remark 2.2. Set

$$\begin{eqnarray}\widetilde{\unicode[STIX]{x1D6F4}}:=\{(A,U)\in \widetilde{Z}\,|\,\text{rank}(A)\leqslant 1\}.\end{eqnarray}$$

The variety $\widetilde{\unicode[STIX]{x1D6F4}}$ is the exceptional locus of $f$ . It is a locally trivial bundle over $G$ with fiber the affine cone over a conic in $\mathbb{P}^{2}$ . Using the isomorphism $\widetilde{Z}=\mathbf{Sym}_{G}^{2}\,{\mathcal{U}}$ , the variety $\widetilde{\unicode[STIX]{x1D6F4}}$ is identified with the locus parametrizing singular symmetric bilinear forms on the fibers of the dual of the tautological rank-2 vector bundle  ${\mathcal{U}}$ . In particular, $\widetilde{\unicode[STIX]{x1D6F4}}$ is a fibration over $G$ in cones over a smooth conic, i.e. $\widetilde{\unicode[STIX]{x1D6F4}}$ is singular only along the zero-section $\widetilde{\unicode[STIX]{x1D6FA}}\simeq G$ of $\mathbf{Sym}_{G}^{2}\,{\mathcal{U}}$ and it has an $A_{1}$ singularity along it.

The following theorem due to Lehn and Sorger [Reference Lehn and SorgerLS06] gives an intrinsic reformulation of the symplectic desingularization $f:\widetilde{Z}\rightarrow Z$ .

Theorem 2.3 [Reference Lehn and SorgerLS06].

Let $p\in \unicode[STIX]{x1D6FA}$ be a singular point of the singular locus $\unicode[STIX]{x1D6F4}\subset K_{v}$ . Then,

  1. (a) [Reference Lehn and SorgerLS06, Théorème 4.5] there is a local analytic isomorphism

    $$\begin{eqnarray}(Z,0)\stackrel{\text{loc}}{\cong }(K_{v},p);\end{eqnarray}$$
  2. (b) [Reference Lehn and SorgerLS06, Théorème 3.1] the resolution $f:\widetilde{Z}\rightarrow Z$ , defined above, coincides with the blow up of $Z$ along its singular locus $\unicode[STIX]{x1D6F4}$ .

In order to discuss the topology of the symplectic desingularization of $f_{v}:\widetilde{K}_{v}\rightarrow K_{v}$ , we are going to describe $f$ in terms of blow ups along smooth subvarieties.

Proposition 2.4. Let $\overline{\unicode[STIX]{x1D6F4}}$ be the strict transform of $\unicode[STIX]{x1D6F4}$ in $Bl_{\unicode[STIX]{x1D6FA}}Z$ .

  1. (1) $\overline{\unicode[STIX]{x1D6F4}}$ is the singular locus of $Bl_{\unicode[STIX]{x1D6FA}}$ , $\overline{\unicode[STIX]{x1D6F4}}$ is smooth and $Bl_{\unicode[STIX]{x1D6FA}}Z$ has an $A_{1}$ singularity along $\overline{\unicode[STIX]{x1D6F4}}$ .

  2. (2) The varieties $Bl_{\widetilde{\unicode[STIX]{x1D6FA}}}Bl_{\unicode[STIX]{x1D6F4}}Z$ and $Bl_{\overline{\unicode[STIX]{x1D6F4}}}Bl_{\unicode[STIX]{x1D6FA}}Z$ are smooth and isomorphic over $Z$ . In particular, the diagram

    where the arrows are blow up maps, is commutative.

Proof. (1) Let $\mathbb{P}(Z):=Z/\mathbb{C}^{\ast }$ be the projectivization of the affine cone Z. As $Z$ is a cone, its blow up $Bl_{\unicode[STIX]{x1D6FA}}Z$ at the origin is the total space of the tautological line bundle over $\mathbb{P}(Z)$ . The singular locus $\unicode[STIX]{x1D6F4}$ of $Z$ is a subcone, hence its strict transform $\overline{\unicode[STIX]{x1D6F4}}=Bl_{\unicode[STIX]{x1D6FA}}\unicode[STIX]{x1D6F4}$ is the total space of the restriction to $\mathbb{P}(\unicode[STIX]{x1D6F4})\subset \mathbb{P}(Z)$ of the tautological line bundle. As $\unicode[STIX]{x1D6F4}\setminus \{0\}$ is smooth, $\mathbb{P}(\unicode[STIX]{x1D6F4})$ is smooth. Moreover, since $Z$ has an $A_{1}$ singularity along $\unicode[STIX]{x1D6F4}\setminus \{0\}$ , the singular locus of $\mathbb{P}(Z)$ is $\mathbb{P}(\unicode[STIX]{x1D6F4})$ , and $\mathbb{P}(Z)$ has an $A_{1}$ singularity along $\mathbb{P}(\unicode[STIX]{x1D6F4})$ . Passing to the total spaces of the tautological line bundles we get item (1).

(2) We only need to show that $Bl_{\widetilde{\unicode[STIX]{x1D6FA}}}Bl_{\unicode[STIX]{x1D6F4}}Z$ and $Bl_{\overline{\unicode[STIX]{x1D6F4}}}Bl_{0}Z$ are isomorphic. By Remark 2.1, $\widetilde{Z}$ is isomorphic to $\mathbf{Sym}_{G}^{2}\,{\mathcal{U}}$ and, by Remark 2.2, $Bl_{\widetilde{\unicode[STIX]{x1D6FA}}}Bl_{\unicode[STIX]{x1D6F4}}Z$ is the blow up of $\mathbf{Sym}_{G}^{2}\,{\mathcal{U}}$ along its zero section. Letting $\mathbb{P}(\operatorname{Sym}_{G}^{2}{\mathcal{U}})$ be the projective bundle associated to $\operatorname{Sym}_{G}^{2}{\mathcal{U}}$ , the blow up $Bl_{\widetilde{\unicode[STIX]{x1D6FA}}}Bl_{\unicode[STIX]{x1D6F4}}Z$ is isomorphic to the total space $T\subset \mathbb{P}(\operatorname{Sym}_{G}^{2}{\mathcal{U}})\times _{G}\mathbf{Sym}_{G}^{2}\,{\mathcal{U}}$ of the tautological line bundle of the projective bundle $\mathbb{P}(\operatorname{Sym}_{G}^{2}{\mathcal{U}})$ . The isomorphism

$$\begin{eqnarray}\mathbf{Sym}_{G}^{2}\,{\mathcal{U}}=\widetilde{Z}:=\{(A,U)\,|\,A(U)=0\}\subset Z\times G\end{eqnarray}$$

also implies

$$\begin{eqnarray}\mathbb{P}(\operatorname{Sym}_{G}^{2}{\mathcal{U}})=\{([A],U)\,|\,A(U)=0\}\subset \mathbb{P}(Z)\times G\end{eqnarray}$$

and, using this identification, we conclude that

$$\begin{eqnarray}Bl_{\widetilde{\unicode[STIX]{x1D6FA}}}Bl_{\unicode[STIX]{x1D6F4}}Z=T=\{([A],B,U)\,|\,A(U)=0,~B\in [A]\}\subset \mathbb{P}(Z)\times Z\times G.\end{eqnarray}$$

On the other side of the diagram, as $Z$ is a cone, its blow up at the origin can explicitly be given as

$$\begin{eqnarray}Bl_{\unicode[STIX]{x1D6FA}}Z=\{([A],B)\,|\,B\in [A]\}\subset \mathbb{P}(Z)\times Z.\end{eqnarray}$$

It remains to show that the map $\unicode[STIX]{x1D709}:Bl_{\widetilde{\unicode[STIX]{x1D6FA}}}Bl_{\unicode[STIX]{x1D6F4}}Z\rightarrow Bl_{\unicode[STIX]{x1D6FA}}Z$ induced by the projection $\unicode[STIX]{x1D70B}_{1,2}:\mathbb{P}(Z)\times Z\times G\rightarrow \mathbb{P}(Z)\times Z$ is the blow up of $\mathbb{P}(Z)\times Z$ along $\overline{\unicode[STIX]{x1D6F4}}$ . Since for $q\in \overline{\unicode[STIX]{x1D6F4}}$ the schematic fiber $\unicode[STIX]{x1D709}^{-1}(q)$ is isomorphic to $\mathbb{P}^{1}$ and $\overline{\unicode[STIX]{x1D6F4}}$ is smooth, the schematic inverse image $\unicode[STIX]{x1D709}^{-1}(\overline{\unicode[STIX]{x1D6F4}})$ is a smooth, hence reduced and irreducible Cartier divisor. By the universal property of blow ups, $\unicode[STIX]{x1D709}$ factors through a proper map $\unicode[STIX]{x1D704}:Bl_{\widetilde{\unicode[STIX]{x1D6FA}}}Bl_{\unicode[STIX]{x1D6F4}}Z\rightarrow Bl_{\overline{\unicode[STIX]{x1D6F4}}}Bl_{\unicode[STIX]{x1D6FA}}Z$ sending $\unicode[STIX]{x1D709}^{-1}(\overline{\unicode[STIX]{x1D6F4}})$ surjectively onto the exceptional divisor of the blow up of $Bl_{\unicode[STIX]{x1D6FA}}Z$ along $\overline{\unicode[STIX]{x1D6F4}}$ . Finally, since $Bl_{\unicode[STIX]{x1D6FA}}Z$ is only singular along $\overline{\unicode[STIX]{x1D6F4}}$ and has an $A_{1}$ singularity along $\overline{\unicode[STIX]{x1D6F4}}$ , the blow up $Bl_{\overline{\unicode[STIX]{x1D6F4}}}Bl_{\unicode[STIX]{x1D6FA}}Z$ is smooth. It follows that $\unicode[STIX]{x1D704}$ is a proper birational map between smooth varieties that does not contract any divisor, therefore $\unicode[STIX]{x1D704}$ is a isomorphism.◻

This proposition also allows us to describe the exceptional loci of the blow up maps appearing in item (2).

Let $\widehat{\unicode[STIX]{x1D6F4}}\subset Bl_{\widetilde{\unicode[STIX]{x1D6FA}}}Bl_{\unicode[STIX]{x1D6F4}}Z$ be the exceptional divisor of $\unicode[STIX]{x1D709}$ , let $\widehat{\unicode[STIX]{x1D6FA}}\subset Bl_{\widetilde{\unicode[STIX]{x1D6FA}}}Bl_{\unicode[STIX]{x1D6F4}}Z$ be the exceptional divisor of $\unicode[STIX]{x1D70C}$ , and recall that $\widetilde{\unicode[STIX]{x1D6FA}}\cong G$ is the inverse image of $\unicode[STIX]{x1D6FA}$ under the resolution $f$ .

Corollary 2.5.

  1. (1) $\widehat{\unicode[STIX]{x1D6F4}}$ is a $\mathbb{P}^{1}$ -bundle over $\overline{\unicode[STIX]{x1D6F4}}$ and $\widehat{\unicode[STIX]{x1D6F4}}=Bl_{\widetilde{\unicode[STIX]{x1D6FA}}}\widetilde{\unicode[STIX]{x1D6F4}}$ .

  2. (2) $\widehat{\unicode[STIX]{x1D6FA}}$ is a $\mathbb{P}^{2}$ -bundle over $\widetilde{\unicode[STIX]{x1D6FA}}$ isomorphic to $\mathbb{P}(\operatorname{Sym}_{G}^{2}{\mathcal{U}})$ .

Proof. (1) By item (1) of Proposition 2.4, the restriction of $\unicode[STIX]{x1D709}$ realizes $\widehat{\unicode[STIX]{x1D6F4}}$ as a $\mathbb{P}^{1}$ -bundle over $\overline{\unicode[STIX]{x1D6F4}}$ . Since $\widehat{\unicode[STIX]{x1D6F4}}$ is also the strict transform of $\widetilde{\unicode[STIX]{x1D6F4}}$ under $\unicode[STIX]{x1D70C}$ and $\widetilde{\unicode[STIX]{x1D6FA}}\subset \widetilde{\unicode[STIX]{x1D6F4}}$ , the restriction of $\unicode[STIX]{x1D70C}$ to $\widehat{\unicode[STIX]{x1D6F4}}$ can be identified with the blow up map of $\widetilde{\unicode[STIX]{x1D6F4}}$ along $\widetilde{\unicode[STIX]{x1D6FA}}$ . As for (2), we can argue as follows. Since $\widetilde{\unicode[STIX]{x1D6FA}}$ is a smooth subvariety of codimension $3$ in the smooth variety $\widetilde{Z}$ , the restriction of $\unicode[STIX]{x1D70C}$ to $\widehat{\unicode[STIX]{x1D6FA}}$ makes it a $\mathbb{P}^{2}$ -bundle over $\widetilde{\unicode[STIX]{x1D6FA}}$ . More precisely, since $\widetilde{\unicode[STIX]{x1D6FA}}$ is the zero section of $\widetilde{Z}=\mathbf{Sym}_{G}^{2}\,{\mathcal{U}}$ (see Remark 2.2), there is an isomorphism $\widehat{\unicode[STIX]{x1D6FA}}\simeq \mathbb{P}(\operatorname{Sym}_{G}^{2}{\mathcal{U}})$ .◻

To compute invariants of $\widetilde{K}_{v}$ we need the following global versions of Proposition 2.4 and Corollary 2.5.

Proposition 2.6. Let $\overline{\unicode[STIX]{x1D6F4}}_{v}$ be the strict transform of $\unicode[STIX]{x1D6F4}_{v}$ in $Bl_{\unicode[STIX]{x1D6FA}_{v}}K_{v}$ .

  1. (1) $\overline{\unicode[STIX]{x1D6F4}}_{v}$ is the singular locus of $Bl_{\unicode[STIX]{x1D6FA}_{v}}K_{v}$ , $\overline{\unicode[STIX]{x1D6F4}}_{v}$ is smooth and $Bl_{\unicode[STIX]{x1D6FA}_{K_{v}}}K_{v}$ has an $A_{1}$ singularity along $\overline{\unicode[STIX]{x1D6F4}}_{v}$ .

  2. (2) The projective varieties $Bl_{\widetilde{\unicode[STIX]{x1D6FA}}_{v}}Bl_{\unicode[STIX]{x1D6F4}_{v}}K_{v}$ and $Bl_{\overline{\unicode[STIX]{x1D6F4}}_{v}}Bl_{\unicode[STIX]{x1D6FA}_{v}}K_{v}$ are smooth and isomorphic over $K_{v}$ . Hence the diagram

    where the arrows are blow ups, is commutative.

Proof. As $\unicode[STIX]{x1D6F4}_{v}\setminus \unicode[STIX]{x1D6FA}_{v}$ is smooth and $K_{v}$ has an $A_{1}$ singularity along $\unicode[STIX]{x1D6F4}_{v}\setminus \unicode[STIX]{x1D6FA}_{v}$ , item (1) follows from Theorem 2.3(a) and Proposition 2.4(1), since the blow up is a local construction. Item (2) holds since item (2) of Proposition 2.4 also implies that the natural birational map between $Bl_{\widetilde{\unicode[STIX]{x1D6FA}}_{v}}Bl_{\unicode[STIX]{x1D6F4}_{v}}K_{v}$ and $Bl_{\overline{\unicode[STIX]{x1D6F4}}_{v}}Bl_{\unicode[STIX]{x1D6FA}_{v}}K_{v}$ is actually an isomorphism.◻

Remark 2.7. Since $\unicode[STIX]{x1D6F4}_{v}$ contains $\unicode[STIX]{x1D6FA}_{v}$ as a closed subscheme, its strict transform $\overline{\unicode[STIX]{x1D6F4}}_{v}$ in $Bl_{\unicode[STIX]{x1D6FA}_{v}}K_{v}$ is isomorphic to the blow up $Bl_{\unicode[STIX]{x1D6FA}_{v}}\unicode[STIX]{x1D6F4}_{v}$ . Recall that $\unicode[STIX]{x1D6F4}_{v}\simeq (A\times A^{\vee })/\pm 1$ , so that its singular locus $\text{Sing}((A\times A^{\vee })/\pm 1)$ is in bijective correspondence with the set of $2$ -torsion points $(A\times A^{\vee })[2]$ of $A\times A^{\vee }$ . It follows that there is a chain of isomorphisms

$$\begin{eqnarray}\overline{\unicode[STIX]{x1D6F4}}_{v}\simeq Bl_{\text{Sing}((A\times A^{\vee })/\pm 1)}((A\times A^{\vee })/\pm 1)\simeq (Bl_{(A\times A^{\vee })[2]}(A\times A^{\vee }))/\pm 1.\end{eqnarray}$$

This also implies that the exceptional divisor of $Bl_{\unicode[STIX]{x1D6FA}_{v}}\unicode[STIX]{x1D6F4}_{v}$ , which is given by the (reduced induced) intersection of the exceptional divisor $\overline{\unicode[STIX]{x1D6FA}}_{v}$ of $Bl_{\unicode[STIX]{x1D6FA}_{v}}K_{v}$ and $\overline{\unicode[STIX]{x1D6F4}}_{v}$ , consists of a union of $256$ disjoint  $\mathbb{P}^{3}$ .

Corollary 2.8. Let $\widehat{\unicode[STIX]{x1D6F4}}_{v}\subset Bl_{\widetilde{\unicode[STIX]{x1D6FA}}_{v}}Bl_{\unicode[STIX]{x1D6F4}_{v}}K_{v}$ be the exceptional divisor of $\unicode[STIX]{x1D709}$ , let $\widehat{\unicode[STIX]{x1D6FA}}_{v}\subset Bl_{\widetilde{\unicode[STIX]{x1D6FA}}_{v}}Bl_{\unicode[STIX]{x1D6F4}_{v}}\widetilde{K}_{v}$ be the exceptional divisor of $\unicode[STIX]{x1D70C}$ , let $\overline{\unicode[STIX]{x1D6FA}}_{v}\subset Bl_{\unicode[STIX]{x1D6FA}_{v}}K_{v}$ be the exceptional divisor of $\unicode[STIX]{x1D702}$ , and, finally, let $\overline{\unicode[STIX]{x1D6FA}}_{v}\cap \overline{\unicode[STIX]{x1D6F4}}_{v}$ denote the intersection of $\overline{\unicode[STIX]{x1D6FA}}_{v}$ and $\overline{\unicode[STIX]{x1D6F4}}_{v}$ with its reduced induced structure.

  1. (1) $\widehat{\unicode[STIX]{x1D6F4}}_{v}$ is a $\mathbb{P}^{1}$ -bundle over $\overline{\unicode[STIX]{x1D6F4}}_{v}$ and $\widehat{\unicode[STIX]{x1D6F4}}_{v}=Bl_{\widetilde{\unicode[STIX]{x1D6FA}}_{v}}\widetilde{\unicode[STIX]{x1D6F4}}_{v}$ .

  2. (2) $\widehat{\unicode[STIX]{x1D6FA}}_{v}$ is a $\mathbb{P}^{2}$ -bundle over $\widetilde{\unicode[STIX]{x1D6FA}}_{v}$ isomorphic to $\mathbb{P}(\operatorname{Sym}_{G}^{2}{\mathcal{U}})$ .

Proof. This follows from item (2) of Theorem 2.3 and Corollary 2.5. ◻

Remark 2.9. The proof of the existence of an isomorphism between the smooth projective varieties $Bl_{\widetilde{\unicode[STIX]{x1D6FA}}_{v}}Bl_{\unicode[STIX]{x1D6F4}_{v}}K_{v}$ and $Bl_{\overline{\unicode[STIX]{x1D6F4}}_{v}}Bl_{\unicode[STIX]{x1D6FA}_{v}}K_{v}$ follows the original strategy used by O’Grady in [Reference O’GradyO’Gr99]. For $v=(2,0,-2)$ , he proved that a symplectic desingularization of $K_{v}$ can be obtained by contracting the strict transform $\widehat{\unicode[STIX]{x1D6FA}}_{v}$ of $\overline{\unicode[STIX]{x1D6FA}}_{v}$ in $Bl_{\overline{\unicode[STIX]{x1D6F4}}_{v}}Bl_{\unicode[STIX]{x1D6FA}_{v}}K_{v}$ . Proposition 2.6 shows, in particular, that O’Grady’s procedure gives a symplectic desingularization of $K_{v}$ that is isomorphic to the Lehn–Sorger desingularization $Bl_{\unicode[STIX]{x1D6F4}_{v}}K_{v}$ . The proof of Proposition 2.6 is elementary because it uses the crucial description, due to Lehn and Sorger, of the analytic type of the singularities appearing in $K_{v}$ .

3 The local covering

This section is devoted to the local description of the double cover, branched along the singular locus, of O’Grady’s singularity.

It is known [Reference Collingwood and McGovernCM93, Corollary 6.1.6] that the fundamental group of the open orbit $\mathfrak{o}(2,2)$ is isomorphic to $\mathbb{Z}/(2)$ . We wish to extend this double cover to a ramified double cover of $\overline{\mathfrak{o}(2,2)}=Z$ .

To this aim, let

$$\begin{eqnarray}W:=\{v\otimes w\,|\,\unicode[STIX]{x1D70E}(v,w)=0\}\subset V\otimes V\quad \text{and}\quad \unicode[STIX]{x1D6E5}=\{v\otimes v\}\subset W\end{eqnarray}$$

be the affine cone over the incidence subvariety

$$\begin{eqnarray}I:=\{([v],[w])\,|\,\unicode[STIX]{x1D70E}(v,w)=0\}\subset \mathbb{P}V\times \mathbb{P}V\subset \mathbb{P}(V\otimes V).\end{eqnarray}$$

Finally denote by

$$\begin{eqnarray}\unicode[STIX]{x1D6E4}\subset W\end{eqnarray}$$

the vertex (i.e. the origin) of $W$ .

Since $I$ is smooth, the vertex $\unicode[STIX]{x1D6E4}$ is also the singular locus of $W$ . Moreover, since $I\subset \mathbb{P}(V\otimes V)$ is projectively normal, $W$ is a normal variety.

Let

$$\begin{eqnarray}\unicode[STIX]{x1D70F}:W\rightarrow W\end{eqnarray}$$

be the involution induced by restricting the linear involution $\unicode[STIX]{x1D70F}_{V\otimes V}$ on $V\otimes V$ that interchanges the two factors.

The following lemma exhibits $W$ as the desired double cover of $Z$ .

Lemma 3.1. The morphism

$$\begin{eqnarray}\displaystyle & & \displaystyle \unicode[STIX]{x1D700}:W\longrightarrow Z,\nonumber\\ \displaystyle & & \displaystyle v\otimes w\longmapsto \unicode[STIX]{x1D70E}(v,\cdot )w+\unicode[STIX]{x1D70E}(w,\cdot )v\nonumber\end{eqnarray}$$

realizes $Z$ as the quotient $W/\unicode[STIX]{x1D70F}$ . In particular, $\unicode[STIX]{x1D700}$ is a finite 2:1 morphism, the ramification locus of $\unicode[STIX]{x1D700}$ is $\unicode[STIX]{x1D6E5}$ and the branch locus of $\unicode[STIX]{x1D700}$ is $\unicode[STIX]{x1D6F4}$ .

Proof. We leave it to the reader to check that $\unicode[STIX]{x1D700}(W)\subset Z$ . For a rank- $2$ endomorphism $A\in Z\setminus \unicode[STIX]{x1D6F4}$ , let us show that $\unicode[STIX]{x1D700}^{-1}(A)$ consists of two points interchanged by $\unicode[STIX]{x1D70F}$ . Let $U\subset V$ be the kernel of $A$ , which is a Lagrangian subspace. As shown in Remark 2.1, $A$ induces a linear map $\unicode[STIX]{x1D711}_{A}\in \operatorname{Hom}(U^{\vee },U)=U\otimes U$ that gives a rank- $2$ bilinear symmetric form on $U^{\vee }$ and, conversely, any symmetric bilinear form on $U^{\vee }$ determines a rank- $2$ endomorphism $A\in Z$ whose kernel is $U$ . A rank- $2$ symmetric bilinear form on $U^{\vee }$ is determined, up to scalars, by two independent distinct isotropic vectors $L_{1}$ and $L_{2}$ , hence by their kernels $\ker (L_{1})\subset U$ and $\ker (L_{2})\subset U$ . Now it suffices to note that, for $v$ and $w$ spanning $U$ and for $A=\unicode[STIX]{x1D700}(v\otimes w)$ , the lines $\ker (L_{1})$ and $\ker (L_{2})$ are the lines generated by $v$ and $w$ .

Since $\ker A$ and $\operatorname{Im}A$ are orthogonal (see Remark 2.1), if $A\in \unicode[STIX]{x1D6F4}$ is a rank- $1$ endomorphism or the $0$ endomorphism, then there exists a unique, up to scalars, $v\in V$ such that $A=\unicode[STIX]{x1D70E}(v,\cdot )v$ . This shows that $\unicode[STIX]{x1D700}^{-1}(A)$ consists of a unique point, which is fixed by $\unicode[STIX]{x1D70F}$ .

To show that $Z\cong W/\unicode[STIX]{x1D70F}$ , note that $\unicode[STIX]{x1D700}$ is $\unicode[STIX]{x1D70F}$ -invariant and its fibers are the orbits of the action of $\unicode[STIX]{x1D70F}$ , hence $\unicode[STIX]{x1D700}$ induces a bijective morphism $W/\unicode[STIX]{x1D70F}\rightarrow Z$ . Since $Z$ is normal (see Criterion $2$ of [Reference HesselinkHes79]), this morphism is an isomorphism.◻

Remark 3.2. Using Lemma 3.1, we may reprove that the fundamental group of $\mathfrak{o}(2,2)=Z\setminus \unicode[STIX]{x1D6F4}$ is isomorphic to $\mathbb{Z}/(2)$ . As $\unicode[STIX]{x1D700}$ is étale on $Z\setminus \unicode[STIX]{x1D6F4}$ , it suffices to show that $\unicode[STIX]{x1D700}^{-1}(Z\setminus \unicode[STIX]{x1D6F4})=W\setminus \unicode[STIX]{x1D6E5}$ is simply connected. One can obtain $W\setminus \unicode[STIX]{x1D6E5}$ from the smooth variety $W\setminus \{0\}$ by removing a codimension- $2$ subvariety, hence there is an isomorphism of fundamental groups $\unicode[STIX]{x1D70B}_{1}(W\setminus \unicode[STIX]{x1D6E5})\simeq \unicode[STIX]{x1D70B}_{1}(W\setminus \{0\})$ . Finally the map $k:W\setminus \{0\}\rightarrow \mathbb{P}(V)$ defined by $k(v\otimes w)=[v]$ is a locally trivial fibration with fiber isomorphic to the complement of $0$ in a three-dimensional vector space. Therefore $k$ has simply connected base and fiber and $\unicode[STIX]{x1D70B}_{1}(W\setminus \{0\})=0$ .

The morphism $\unicode[STIX]{x1D700}$ induces double coverings of the varieties $\widetilde{Z}=Bl_{\unicode[STIX]{x1D6F4}}Z$ , $Bl_{\unicode[STIX]{x1D6FA}}Z$ and $Bl_{\widetilde{\unicode[STIX]{x1D6FA}}}Bl_{\unicode[STIX]{x1D6F4}}Z=Bl_{\overline{\unicode[STIX]{x1D6F4}}}Bl_{\unicode[STIX]{x1D6FA}}$ . The following corollary discusses the case of $Bl_{\unicode[STIX]{x1D6FA}}Z$ .

Corollary 3.3. The morphism $\unicode[STIX]{x1D700}$ lifts to a finite 2:1 morphism

$$\begin{eqnarray}\overline{\unicode[STIX]{x1D700}}:Bl_{\unicode[STIX]{x1D6E4}}W\rightarrow Bl_{\unicode[STIX]{x1D6FA}}Z,\end{eqnarray}$$

whose branch locus is the strict transform $\overline{\unicode[STIX]{x1D6F4}}$ of $\unicode[STIX]{x1D6F4}$ in $Bl_{\unicode[STIX]{x1D6FA}}Z$ .

Proof. The morphism $\unicode[STIX]{x1D700}$ is the restriction to $W$ of the linear map

$$\begin{eqnarray}\unicode[STIX]{x1D700}_{V\otimes V}:V\otimes V\rightarrow \mathfrak{sp}(V)\end{eqnarray}$$

sending $v\otimes w$ to $\unicode[STIX]{x1D70E}(v,\cdot )w+\unicode[STIX]{x1D70E}(w,\cdot )v$ for any $v\otimes w\subset V\otimes V$ . As $\ker \unicode[STIX]{x1D700}_{V\otimes V}\cap W=0$ , the map $\unicode[STIX]{x1D700}$ induces a morphism $\mathbb{P}(\unicode[STIX]{x1D700}):I\rightarrow \mathbb{P}(Z)$ between the projectivization of $W$ and $Z$ . There are the identifications

$$\begin{eqnarray}Bl_{\unicode[STIX]{x1D6E4}}W=Bl_{0}W=\{(\mathbb{C}\unicode[STIX]{x1D6FC},v\otimes w)\in I\times W\,|\,v\otimes w\in \mathbb{C}\unicode[STIX]{x1D6FC}\}\end{eqnarray}$$

and

$$\begin{eqnarray}Bl_{\unicode[STIX]{x1D6FA}}Z=Bl_{0}Z=\{(\mathbb{C}A,B)\in \mathbb{P}(Z)\times Z\,|\,B\in \mathbb{C}A\}.\end{eqnarray}$$

It follows that $\mathbb{P}(\unicode[STIX]{x1D700})\times \unicode[STIX]{x1D700}$ restricts to a map $\overline{\unicode[STIX]{x1D700}}:Bl_{\{0\}}W\rightarrow Bl_{\unicode[STIX]{x1D6FA}}Z$ and, by Lemma 3.1, $\overline{\unicode[STIX]{x1D700}}$ is a finite $2:1$ map whose branch locus is

$$\begin{eqnarray}\overline{\unicode[STIX]{x1D6F4}}_{W}=Bl_{\unicode[STIX]{x1D6FA}}\unicode[STIX]{x1D6F4}=Bl_{0}\unicode[STIX]{x1D6F4}=\{(\mathbb{C}A,B)\in \mathbb{P}(\unicode[STIX]{x1D6F4})\times \unicode[STIX]{x1D6F4}\,|\,B\in \mathbb{C}A\}.\Box\end{eqnarray}$$

Remark 3.4. Since $W$ is the cone over a smooth variety, both its blow up at the origin $Bl_{\unicode[STIX]{x1D6E4}}W$ and the exceptional divisor $\overline{\unicode[STIX]{x1D6E4}}\subset Bl_{\unicode[STIX]{x1D6E4}}W$ are smooth. Finally the strict transform $\overline{\unicode[STIX]{x1D6E5}}$ of $\unicode[STIX]{x1D6E5}$ in $Bl_{\unicode[STIX]{x1D6E4}}W$ is isomorphic to $Bl_{\unicode[STIX]{x1D6E4}}\unicode[STIX]{x1D6E5}$ and, since $\unicode[STIX]{x1D6E5}$ is the cone over a smooth variety, also $\overline{\unicode[STIX]{x1D6E5}}$ is smooth.

The following corollary treats the case of the induced double cover of $\widetilde{Z}$ .

Corollary 3.5. Let $\unicode[STIX]{x1D70B}:SC_{G}\,{\mathcal{U}}^{\otimes 2}\rightarrow G$ be the relative affine Segre cone parametrizing decomposable tensors in the total space of the rank- $4$ vector bundle ${\mathcal{U}}^{\otimes 2}$ .

  1. (1) $SC_{G}\,{\mathcal{U}}^{\otimes 2}$ is isomorphic to $Bl_{\unicode[STIX]{x1D6E5}}W$ .

  2. (2) Using this identification, the map

    $$\begin{eqnarray}\widetilde{\unicode[STIX]{x1D700}}:Bl_{\unicode[STIX]{x1D6E5}}W~(=SC_{G}\,{\mathcal{U}}^{\otimes 2})\rightarrow \widetilde{Z}~(=\mathbf{Sym}_{G}^{2}\,{\mathcal{U}}),\end{eqnarray}$$
    induced by symmetrization on the fibers, is a finite 2:1 morphism lifting $\unicode[STIX]{x1D700}$ , whose branch locus is $\widetilde{\unicode[STIX]{x1D6F4}}$ .

Proof. (1) By definition of fiber product, $W\times _{Z}\widetilde{Z}$ is equal to

$$\begin{eqnarray}\{(v\otimes w,A,U)\in W\times Z\times G\,|\,\unicode[STIX]{x1D700}(v\otimes w)=A~\text{and}~v,w\in U\}\end{eqnarray}$$

and, by Lemma 3.1, the fiber over $U$ of the projection $\unicode[STIX]{x1D70B}_{G}:W\times _{Z}\widetilde{Z}\rightarrow G$ is naturally isomorphic to the variety $SC\,U^{\otimes 2}$ of decomposable tensors in $U\otimes U$ . It follows that $SC_{G}\,{\mathcal{U}}^{\otimes 2}$ is isomorphic to $W\times _{Z}\widetilde{Z}$ .

Let us show that $W\times _{Z}\widetilde{Z}$ has a birational morphism to $Bl_{\unicode[STIX]{x1D6E5}}W$ . Let $\unicode[STIX]{x1D70B}_{W}:W\times _{Z}\widetilde{Z}\rightarrow W$ be the projection. By the universal property of blow ups, it will suffice to show that the schematic inverse image $\unicode[STIX]{x1D70B}_{W}^{-1}(\unicode[STIX]{x1D6E5})$ is a Cartier divisor.

For $U\in G$ , the projection $\unicode[STIX]{x1D70B}_{W}$ sends the fiber $\unicode[STIX]{x1D70B}_{G}^{-1}(U)$ isomorphically onto $SC\,U^{\otimes 2}$ . Hence the schematic intersection $\unicode[STIX]{x1D70B}_{W}^{-1}(\unicode[STIX]{x1D6E5})\cap \unicode[STIX]{x1D70B}_{G}^{-1}(U)$ is isomorphic to the schematic intersection $\unicode[STIX]{x1D6E5}\cap SC\,U^{\otimes 2}$ , i.e. the reduced cone over a smooth conic $C\subset \mathbb{P}(U\otimes U)$ parametrizing symmetric decomposable tensors in $U\otimes U$ . As, on varying $U\in G$ , the intersections $\unicode[STIX]{x1D70B}_{W}^{-1}(\unicode[STIX]{x1D6E5})\cap \unicode[STIX]{x1D70B}_{G}^{-1}(U)$ form a locally trivial family over $G$ , the family $\unicode[STIX]{x1D70B}_{W}^{-1}(\unicode[STIX]{x1D6E5})\rightarrow G$ is locally trivial. Finally, as the cone over $C\subset \mathbb{P}(U\otimes U)$ is a Cartier divisor in the variety of decomposable tensors of $U\otimes U$ , the scheme $\unicode[STIX]{x1D70B}_{W}^{-1}(\unicode[STIX]{x1D6E5})$ is a Cartier divisor in $W\times _{Z}\widetilde{Z}$ .

On the other hand, $Bl_{\unicode[STIX]{x1D6E5}}W$ has a regular birational morphism to $W\times _{Z}\widetilde{Z}$ inverting the previous birational morphism.

This will follow if we prove that the ideal of $\unicode[STIX]{x1D700}^{-1}(\unicode[STIX]{x1D6F4})$ in $W$ is the square $I_{\unicode[STIX]{x1D6E5}}^{2}$ of the ideal of $\unicode[STIX]{x1D6E5}$ . In fact, in this case, the blow up $Bl_{\unicode[STIX]{x1D6E5}}W$ equals the blow up $Bl_{\unicode[STIX]{x1D700}^{-1}(\unicode[STIX]{x1D6F4})}W$ ; hence the schematic inverse image of $\unicode[STIX]{x1D6F4}$ in $Bl_{\unicode[STIX]{x1D6E5}}W$ is a Cartier divisor. Therefore, as $\widetilde{Z}=Bl_{\unicode[STIX]{x1D6F4}}Z$ , by the universal property of blow ups, we can conclude that there exists a commutative diagram

inducing the desired birational regular morphism from $Bl_{\unicode[STIX]{x1D6E5}}W$ to $W\times _{Z}\widetilde{Z}$ .

To determine the ideals of $\unicode[STIX]{x1D700}^{-1}(\unicode[STIX]{x1D6F4})$ and $\unicode[STIX]{x1D6E5}$ in $W$ , we recall that the involution $\unicode[STIX]{x1D70F}$ is the restriction of the linear involution $\unicode[STIX]{x1D70F}_{V\otimes V}$ on $V\otimes V$ which can be interpreted as the transposition on $4\times 4$ matrices if we choose a basis for $V$ . Moreover, the ideal of $\unicode[STIX]{x1D6E5}$ in $W$ is generated by the restrictions of the linear antiinvariant functions on $V\otimes V$ (this already holds for the ideal of $\unicode[STIX]{x1D6E5}$ in the affine cone over the Segre variety $\mathbb{P}(V)\times \mathbb{P}(V)\subset \mathbb{P}(V\otimes V)$ ). Hence, $I_{\unicode[STIX]{x1D6E5}}^{2}$ is generated by restrictions of products of pairs of linear antiinvariant functions on $V\otimes V$ and any such product comes from a function on the quotient $(V\otimes V)/\unicode[STIX]{x1D70F}_{V\otimes V}$ vanishing along the branch locus $B$ . Since $B$ contains the branch locus $\unicode[STIX]{x1D6F4}$ of $W/\unicode[STIX]{x1D70F}$ , we conclude that the ideal of $\unicode[STIX]{x1D700}^{-1}(\unicode[STIX]{x1D6F4})$ contains $I_{\unicode[STIX]{x1D6E5}}^{2}$ . Equality holds because $W\setminus \{0\}$ is smooth and the fixed locus $\unicode[STIX]{x1D6E5}\setminus \{0\}$ has codimension $2$ , hence $\unicode[STIX]{x1D700}^{-1}(\unicode[STIX]{x1D6F4})$ equals the subscheme $\unicode[STIX]{x1D6E5}_{2}$ defined by $I_{\unicode[STIX]{x1D6E5}}^{2}$ outside the origin. As $\unicode[STIX]{x1D6E5}_{2}$ is a subcone of $W$ , it is the closure of $\unicode[STIX]{x1D6E5}_{2}\setminus \{0\}$ , therefore it is a closed subscheme of $\unicode[STIX]{x1D700}^{-1}(\unicode[STIX]{x1D6F4})$ .

(2) The existence of the regular morphism $\widetilde{\unicode[STIX]{x1D700}}$ lifting $\unicode[STIX]{x1D700}$ follows from (1). The branch locus of $\tilde{\unicode[STIX]{x1D700}}$ is $\widetilde{\unicode[STIX]{x1D6F4}}$ because, by our description of $\tilde{\unicode[STIX]{x1D700}}$ , it parametrizes singular bilinear symmetric tensors (see Remark 2.2).◻

Corollary 3.5 also allows us to describe the singularities of the exceptional divisor $\widetilde{\unicode[STIX]{x1D6E5}}$ of the blow up $Bl_{\unicode[STIX]{x1D6E5}}W$ of $W$ along $\unicode[STIX]{x1D6E5}$ .

Remark 3.6. We have that $Bl_{\unicode[STIX]{x1D6E5}}W\simeq SC_{G}\,{\mathcal{U}}^{\otimes 2}$ is a locally trivial bundle over $G$ with fiber the affine cone over a smooth quadric in $\mathbb{P}^{3}$ . Hence it is smooth outside the zero section $\widetilde{\unicode[STIX]{x1D6E4}}$ and any point of $\widetilde{\unicode[STIX]{x1D6E4}}$ has a neighborhood isomorphic to the product of the affine cone over a smooth quadric and a smooth three-dimensional variety. As $\unicode[STIX]{x1D700}$ and $\widetilde{\unicode[STIX]{x1D700}}$ are finite, the morphism $\widetilde{\unicode[STIX]{x1D700}}$ sends the exceptional divisor $\widetilde{\unicode[STIX]{x1D6E5}}\subset Bl_{\unicode[STIX]{x1D6E5}}W$ onto the exceptional divisor $\widetilde{\unicode[STIX]{x1D6F4}}\subset \widetilde{Z}$ . By the definition of $\widetilde{\unicode[STIX]{x1D700}}$ in item (2) of Corollary 3.5, the divisor $\widetilde{\unicode[STIX]{x1D6E5}}$ parametrizes symmetric decomposable tensors in the fibers of $\unicode[STIX]{x1D70B}_{G}:SC_{G}\,{\mathcal{U}}^{\otimes 2}\rightarrow G$ , hence it is a locally trivial bundle with fiber the affine cone over a smooth conic. Therefore it is smooth outside $\widetilde{\unicode[STIX]{x1D6E4}}$ and has an $A_{1}$ singularity along $\widetilde{\unicode[STIX]{x1D6E4}}$ .

The following corollary completes the picture of the double covering induced by $\unicode[STIX]{x1D700}$ in the local case.

Corollary 3.7.

  1. (1) There exist finite degree- $2$ morphisms $\widehat{\unicode[STIX]{x1D700}}_{1}:Bl_{\widetilde{\unicode[STIX]{x1D6E4}}}Bl_{\unicode[STIX]{x1D6E5}}W\rightarrow Bl_{\widetilde{\unicode[STIX]{x1D6FA}}}Bl_{\unicode[STIX]{x1D6F4}}Z$ and $\widehat{\unicode[STIX]{x1D700}}_{2}:Bl_{\overline{\unicode[STIX]{x1D6E5}}}Bl_{\unicode[STIX]{x1D6E4}}W\rightarrow Bl_{\overline{\unicode[STIX]{x1D6F4}}}Bl_{\unicode[STIX]{x1D6FA}}Z$ , lifting $\widetilde{\unicode[STIX]{x1D700}}$ and $\overline{\unicode[STIX]{x1D700}}$ , whose branch loci are the strict transform of $\widetilde{\unicode[STIX]{x1D6F4}}$ in $Bl_{\widetilde{\unicode[STIX]{x1D6FA}}}Bl_{\unicode[STIX]{x1D6F4}}Z$ and the exceptional divisor $\widehat{\unicode[STIX]{x1D6F4}}$ of $Bl_{\overline{\unicode[STIX]{x1D6F4}}}Bl_{\unicode[STIX]{x1D6FA}}Z$ , respectively.

  2. (2) The varieties $Bl_{\widetilde{\unicode[STIX]{x1D6E4}}}Bl_{\unicode[STIX]{x1D6E5}}W$ and $Bl_{\overline{\unicode[STIX]{x1D6E5}}}Bl_{\unicode[STIX]{x1D6E4}}W$ are smooth and isomorphic over $W$ . Hence there exists a commutative diagram

    where the diagonal arrows are blow ups.

Proof. (1) Recall that $Bl_{\unicode[STIX]{x1D6E5}}W\simeq SC_{G}\,{\mathcal{U}}^{\otimes 2}$ and $Bl_{\unicode[STIX]{x1D6F4}}Z\simeq \mathbf{Sym}_{G}^{2}\,{\mathcal{U}}$ are locally trivial bundles over $G$ , and $\widetilde{\unicode[STIX]{x1D6E4}}$ and $\widetilde{\unicode[STIX]{x1D6FA}}$ are their respective zero sections. As $\widetilde{\unicode[STIX]{x1D700}}:Bl_{\unicode[STIX]{x1D6E5}}W\rightarrow Bl_{\unicode[STIX]{x1D6F4}}Z$ is a morphism over $G$ , the existence of $\widehat{\unicode[STIX]{x1D700}}_{1}:Bl_{\widetilde{\unicode[STIX]{x1D6E4}}}Bl_{\unicode[STIX]{x1D6E5}}W\rightarrow Bl_{\widetilde{\unicode[STIX]{x1D6FA}}}Bl_{\unicode[STIX]{x1D6F4}}Z$ branched over the strict transform of $\widetilde{\unicode[STIX]{x1D6F4}}$ in $Bl_{\widetilde{\unicode[STIX]{x1D6FA}}}Bl_{\unicode[STIX]{x1D6F4}}Z$ follows from the existence of a commutative diagram of the form

where $U$ is a two-dimensional vector space, $SC\,U^{\otimes 2}\subset U\otimes U$ is the affine cone, parametrizing decomposable tensors, over the Segre variety $\mathbb{P}(U)\times \mathbb{P}(U)$ , the vertical arrows are blow ups, and the horizontal arrows are induced by symmetrization (hence their branch locus parametrizes singular symmetric tensors).

By Corollary 3.3, the branch locus of the finite $2:1$ morphism $\overline{\unicode[STIX]{x1D700}}:Bl_{\unicode[STIX]{x1D6E4}}W\rightarrow Bl_{\unicode[STIX]{x1D6FA}}Z$ is the singular locus $\overline{\unicode[STIX]{x1D6F4}}$ of $Bl_{\unicode[STIX]{x1D6FA}}Z$ . By item (1) of Proposition 2.4, $Bl_{\unicode[STIX]{x1D6FA}}Z$ has an $A_{1}$ singularity along $\overline{\unicode[STIX]{x1D6F4}}$ and this suffices to imply the existence of the desired finite $2:1$ morphism $\widehat{\unicode[STIX]{x1D700}}_{2}:Bl_{\overline{\unicode[STIX]{x1D6E5}}}Bl_{\unicode[STIX]{x1D6E4}}W\rightarrow Bl_{\overline{\unicode[STIX]{x1D6F4}}}Bl_{\unicode[STIX]{x1D6FA}}Z$ whose branch locus is the exceptional divisor $\widehat{\unicode[STIX]{x1D6F4}}$ of $Bl_{\overline{\unicode[STIX]{x1D6F4}}}Bl_{\unicode[STIX]{x1D6FA}}Z$ .

(2) The smoothness of $Bl_{\widetilde{\unicode[STIX]{x1D6E4}}}Bl_{\unicode[STIX]{x1D6E5}}W$ and $Bl_{\overline{\unicode[STIX]{x1D6E5}}}Bl_{\unicode[STIX]{x1D6E4}}W$ follows from Remark 3.6 and Remark 3.4, respectively.

It remains to show that the natural birational map $j:Bl_{\widetilde{\unicode[STIX]{x1D6E4}}}Bl_{\unicode[STIX]{x1D6E5}}W{\dashrightarrow}Bl_{\overline{\unicode[STIX]{x1D6E5}}}Bl_{\unicode[STIX]{x1D6E4}}W$ extends to a biregular morphism. Using the identification $Bl_{\widetilde{\unicode[STIX]{x1D6FA}}}Bl_{\unicode[STIX]{x1D6F4}}Z=Bl_{\overline{\unicode[STIX]{x1D6F4}}}Bl_{\unicode[STIX]{x1D6FA}}$ , $\widehat{\unicode[STIX]{x1D700}}_{1}$ and $\widehat{\unicode[STIX]{x1D700}}_{2}$ may be seen as finite covers of $Bl_{\widetilde{\unicode[STIX]{x1D6FA}}}Bl_{\unicode[STIX]{x1D6F4}}Z$ and we have an equality of rational maps $\unicode[STIX]{x1D700}_{2}\circ j=\unicode[STIX]{x1D700}_{1}$ . It follows that the closure of the graph of $j$ is contained in the fiber product $Bl_{\widetilde{\unicode[STIX]{x1D6E4}}}Bl_{\unicode[STIX]{x1D6E5}}W\times _{Bl_{\widetilde{\unicode[STIX]{x1D6FA}}}Bl_{\unicode[STIX]{x1D6F4}}Z}Bl_{\overline{\unicode[STIX]{x1D6E5}}}Bl_{\unicode[STIX]{x1D6E4}}W$ . As $\widehat{\unicode[STIX]{x1D700}}_{1}$ and $\widehat{\unicode[STIX]{x1D700}}_{2}$ are finite, the closure of the graph of $j$ is finite and generically injective on the smooth factors $Bl_{\widetilde{\unicode[STIX]{x1D6E4}}}Bl_{\unicode[STIX]{x1D6E5}}W$ and $Bl_{\overline{\unicode[STIX]{x1D6E5}}}Bl_{\unicode[STIX]{x1D6E4}}W$ . By Zariski’s main theorem it is the graph of an isomorphism extending  $j$ .

The commutativity of the diagram holds because all maps are regular and commutativity is trivial on open dense subsets. ◻

In the final remark of this section we discuss the behavior of the restriction of the morphisms appearing in the diagram in item (2) of Corollary 3.7 to the divisors appearing over $\unicode[STIX]{x1D6FA}$ . Since this remark will not be used in the rest of the paper, some of the computations are left to the reader.

Remark 3.8. Let $\widehat{\unicode[STIX]{x1D6E4}}$ be the exceptional divisor of the blow up of $Bl_{\unicode[STIX]{x1D6E5}}W$ along its singular locus $\widetilde{\unicode[STIX]{x1D6E4}}$ (see Remark 3.6). By restricting the morphisms in the upper part of the diagram in Corollary 3.7, we get the following diagram.

As $\widetilde{\unicode[STIX]{x1D6FA}}$ is contained in the branch locus of $\widetilde{\unicode[STIX]{x1D700}}$ , the morphism $a_{4}$ is an isomorphism and $\widetilde{\unicode[STIX]{x1D6E4}}$ and $\widetilde{\unicode[STIX]{x1D6FA}}$ are isomorphic to $G$ . By item (1) of Corollary 3.5, the exceptional divisor $\widehat{\unicode[STIX]{x1D6E4}}$ has a natural identification with $\mathbb{P}({\mathcal{U}})\times _{G}\mathbb{P}({\mathcal{U}})$ and $a_{1}$ is the natural fibration over $G$ . Analogously, by Remark 2.1, the divisor $\widehat{\unicode[STIX]{x1D6FA}}$ is identified with $\mathbb{P}(\operatorname{Sym}^{2}{\mathcal{U}})=\mathbb{P}(\widetilde{Z})$ and $a_{6}$ is the fibration over  $G$ .

The restriction $a_{4}:\mathbb{P}({\mathcal{U}})\times _{G}\mathbb{P}({\mathcal{U}})\rightarrow \mathbb{P}(\operatorname{Sym}^{2}{\mathcal{U}})$ of $\widehat{\unicode[STIX]{x1D700}}_{1,v}$ is the natural $2:1$ morphism.

The birational morphism $a_{2}:\mathbb{P}({\mathcal{U}})\times _{G}\mathbb{P}({\mathcal{U}})\rightarrow I\subset \mathbb{P}(V)\times \mathbb{P}(V)$ is induced by composing with the natural morphism $\mathbb{P}({\mathcal{U}})\rightarrow \mathbb{P}(V)$ .

The birational morphism $a_{7}:\mathbb{P}(\widetilde{Z})\rightarrow Z$ is induced by $f$ , and finally the finite $2:1$ morphism $a_{5}:I\rightarrow \overline{\unicode[STIX]{x1D6FA}}=\mathbb{P}(Z)$ is the map $\mathbb{P}(\unicode[STIX]{x1D700})$ obtained from $\unicode[STIX]{x1D700}$ by projectivization (see the proof of Corollary 3.3).

4 The global covering

In this section we globalize the local double coverings of Lemma 3.7.Footnote 2 Our starting point is the following result contained in [Reference RapagnettaRap04] and [Reference Perego and RapagnettaPR14]. Keeping the notation as above, let $\widetilde{\unicode[STIX]{x1D6F4}}_{v}\subset \widetilde{K}_{v}$ be the exceptional divisor of the blow up $\widetilde{K}_{v}=Bl_{\unicode[STIX]{x1D6F4}}K_{v}\rightarrow K_{v}$ .

Theorem 4.1 [Reference RapagnettaRap04, Theorem 3.3.1].

The class of $\widetilde{\unicode[STIX]{x1D6F4}}_{v}$ in the Picard group $\text{Pic}(\widetilde{K}_{v})$ of $\widetilde{K}_{v}$ is divisible by two.

Proof. The case of $\widetilde{K}_{(2,0,-2)}$ is dealt in [Reference RapagnettaRap04, Theorem 3.3.1]. The general case follows from [Reference Perego and RapagnettaPR14, Theorem 3.1 and Remark 3.4].◻

As the Picard group of the IHS manifold $\widetilde{K}_{v}$ is torsion-free, there exists a unique normal projective variety $\widetilde{Y}_{v}$ equipped with a double cover $\widetilde{\unicode[STIX]{x1D700}}_{v}:\widetilde{Y}_{v}\rightarrow \widetilde{K}_{v}$ branched over $\widetilde{\unicode[STIX]{x1D6F4}}_{v}$ . This double cover allows us to construct the global analogue of the morphism $\unicode[STIX]{x1D700}$ of Lemma 3.7.

Theorem 4.2. There exists a unique normal projective variety $Y_{v}$ equipped with a finite 2:1 morphism $\unicode[STIX]{x1D700}_{v}:Y_{v}\rightarrow K_{v}$ whose branch locus is $\unicode[STIX]{x1D6F4}_{v}$ . The ramified double cover induced by $\unicode[STIX]{x1D700}_{v}$ on a small analytic neighborhood of a point of $\unicode[STIX]{x1D6FA}_{v}$ is isomorphic to the ramified double cover induced by $\unicode[STIX]{x1D700}:W\rightarrow Z$ on a small analytic neighborhood of a point of $\unicode[STIX]{x1D6FA}$ in $Z$ .

Proof. For any $p\in \unicode[STIX]{x1D6FA}_{v}$ there exists a small analytic neighborhood $U_{p,v}$ of $p\in K_{v}$ that is biholomorphic to the intersection of $Z$ with an open ball. Hence, for any $p\in \unicode[STIX]{x1D6FA}_{v}$ there exists a proper complex analytic space $Y_{p,v}$ and a finite $2:1$ morphism $\unicode[STIX]{x1D700}_{p,v}:Y_{p,v}\rightarrow U_{p,v}$ branched along $U_{p,v}\cap \unicode[STIX]{x1D6F4}_{v}$ , which is obtained by restricting $\unicode[STIX]{x1D700}$ .

On the other hand, there exists an analytic manifold $Y_{v}^{o}$ and a finite $2:1$ morphism $\unicode[STIX]{x1D700}_{v}^{o}:Y_{v}^{o}\rightarrow K_{v}\setminus \unicode[STIX]{x1D6FA}_{v}$ branched along $\unicode[STIX]{x1D6F4}_{v}\setminus \unicode[STIX]{x1D6FA}_{v}$ . To see this, first of all note that by restricting $\widetilde{\unicode[STIX]{x1D700}}_{v}:\widetilde{Y}_{v}\rightarrow \widetilde{K}_{v}$ we get a double covering of $\widetilde{K}_{v}\setminus \widetilde{\unicode[STIX]{x1D6FA}}_{v}$ branched along $\widetilde{\unicode[STIX]{x1D6F4}}_{v}\setminus \widetilde{\unicode[STIX]{x1D6FA}}_{v}$ . Since $K_{v}$ has an $A_{1}$ singularity along $\unicode[STIX]{x1D6F4}_{v}$ , the exceptional divisor $\widetilde{\unicode[STIX]{x1D6F4}}_{v}\setminus \widetilde{\unicode[STIX]{x1D6FA}}$ is a $\mathbb{P}^{1}$ -bundle whose normal bundle has degree $-2$ on the fibers. It follows that $\widetilde{\unicode[STIX]{x1D700}}_{v}^{-1}(\widetilde{\unicode[STIX]{x1D6F4}}_{v}\setminus \widetilde{\unicode[STIX]{x1D6FA}})$ is a $\mathbb{P}^{1}$ -bundle whose normal bundle has degree $-1$ on the fibers. By Nakano’s theorem (see [Reference NakanoNak71] and [Reference Fujiki and NakanoFN71/72]), $\widetilde{\unicode[STIX]{x1D700}}_{v}^{-1}(\widetilde{K}_{v}\setminus \widetilde{\unicode[STIX]{x1D6FA}})$ is the blow up of a complex manifold $Y_{v}^{o}$ along a submanifold isomorphic to $\unicode[STIX]{x1D6F4}_{v}\setminus \unicode[STIX]{x1D6FA}_{v}$ . Moreover, since $f_{v}\circ \widetilde{\unicode[STIX]{x1D700}}_{v}:\widetilde{Y}_{v}\rightarrow K_{v}$ is constant on the fibers of the $\mathbb{P}^{1}$ bundle $\widetilde{\unicode[STIX]{x1D700}}_{v}^{-1}(\widetilde{\unicode[STIX]{x1D6F4}}_{v}\setminus \widetilde{\unicode[STIX]{x1D6FA}})$ , it induces the desired finite $2:1$ morphism $\unicode[STIX]{x1D700}_{v}^{o}:Y_{v}^{o}\rightarrow K_{v}\setminus \unicode[STIX]{x1D6FA}_{v}$ .

To yield the existence of $\unicode[STIX]{x1D700}_{v}:Y_{v}\rightarrow K_{v}$ , it will suffice to prove that $\unicode[STIX]{x1D700}_{p,v}$ and $\unicode[STIX]{x1D700}_{v}^{o}$ induce isomorphic double covers on $U_{p}\setminus \{p\}$ so that they can be glued to get $\unicode[STIX]{x1D700}_{v}$ . Recall from Remark 3.2 that the fundamental group of $Z\setminus \unicode[STIX]{x1D6F4}$ is $\mathbb{Z}/2\mathbb{Z}$ . Since the same holds for $U_{p,v}\setminus \unicode[STIX]{x1D6F4}$ , the étale double covers induced by $\unicode[STIX]{x1D700}_{p,v}$ and $\unicode[STIX]{x1D700}_{v}^{o}$ on $U_{p,v}\setminus \unicode[STIX]{x1D6F4}$ are isomorphic. The closure of the graph of this isomorphism in the fiber product $\unicode[STIX]{x1D700}_{v}^{o-1}(U_{p,v}\setminus \{p\})\times _{U_{p,v}\setminus \{p\}}\unicode[STIX]{x1D700}_{p,v}^{-1}(U_{p}\setminus \{p\})$ is finite and bimeromorphic on the manifolds $\unicode[STIX]{x1D700}_{v}^{o-1}(U_{p,v}\setminus \{p\})$ and $\unicode[STIX]{x1D700}_{p,v}^{-1}(U_{p}\setminus \{p\})$ , hence it is the graph of an isomorphism of double covers $K_{v}$ .

The glued complex analytic space $Y_{v}$ is also projective as a consequence of GAGA’s principles [Reference GrothendieckGro63, Corollary 4.6], since it has a finite proper map to a projective variety. Finally $Y_{v}$ is normal since $W$ is normal and since the normality of a complex variety may be checked on the associated complex analytic space [Reference GrothendieckGro63, Proposition 2.1].

To prove uniqueness of $\unicode[STIX]{x1D700}_{v}$ , let $\unicode[STIX]{x1D700}_{v}^{\prime }:Y_{v}^{\prime }\rightarrow K_{v}$ be a finite $2:1$ morphism branched over $\unicode[STIX]{x1D6F4}_{v}$ such that $Y_{v}^{\prime }$ is normal. In this case $Y_{v}\setminus \unicode[STIX]{x1D700}_{v}^{-1}(\unicode[STIX]{x1D6F4}_{v})$ and $Y_{v}^{\prime }\setminus \unicode[STIX]{x1D700}_{v}^{\prime -1}(\unicode[STIX]{x1D6F4}_{v})$ are algebraic proper étale double covers of $K_{v}\setminus \unicode[STIX]{x1D6F4}_{v}=\widetilde{K}_{v}\setminus \widetilde{\unicode[STIX]{x1D6F4}}_{v}$ . Any such cover is determined by a $2$ -torsion point in the Picard group $\text{Pic}(\widetilde{K}_{v}\setminus \widetilde{\unicode[STIX]{x1D6F4}}_{v})$ and a nowhere vanishing section (unique up to scalars) of the trivial line bundle. As $\widetilde{\unicode[STIX]{x1D6F4}}_{v}$ is irreducible and its class is divisible by 2 in the free group $\text{Pic}(\widetilde{K}_{v})$ , there exists a unique nontrivial $2$ -torsion point in $\text{Pic}(\widetilde{K}_{v}\setminus \widetilde{\unicode[STIX]{x1D6F4}}_{v})$ . Moreover, as $K_{v}$ is normal and $\unicode[STIX]{x1D6F4}_{v}$ has codimension 2 in $K_{v}$ , a regular function on $K_{v}\setminus \unicode[STIX]{x1D6F4}_{v}$ extends to the projective variety $K_{v}$ and therefore it is constant. It follows that $Y_{v}\setminus \unicode[STIX]{x1D700}_{v}^{-1}(\unicode[STIX]{x1D6F4}_{v})$ and $Y_{v}^{\prime }\setminus \unicode[STIX]{x1D700}_{v}^{\prime -1}(\unicode[STIX]{x1D6F4}_{v})$ are isomorphic étale double covers of $K_{v}\setminus \unicode[STIX]{x1D6F4}_{v}$ .

Repeating the argument in the final part of the proof of the existence, the closure of the graph of this isomorphism in the fiber product $Y_{v}\times _{K_{v}}Y_{v}^{\prime }$ is finite and birational over the normal varieties $Y_{v}$ and $Y_{v}^{\prime }$ , hence it is the graph of an isomorphism of double covers. The local characterization of $\unicode[STIX]{x1D700}$ near points of $\unicode[STIX]{x1D6FA}_{v}$ holds by construction.◻

Theorem 4.2 allows one to prove a straightforward global version of Lemma 3.7. Let $\unicode[STIX]{x1D6E5}_{v}\subset Y_{v}$ be the ramification locus (with the reduced induced structure) of $\unicode[STIX]{x1D700}_{v}$ and let $\unicode[STIX]{x1D6E4}_{v}$ be the singular locus (consisting of 256 points) of $Y_{v}$ . Denote by $\widetilde{\unicode[STIX]{x1D6E4}}_{v}$ the inverse image with reduced structure of $\unicode[STIX]{x1D6E4}_{v}$ in $Bl_{\unicode[STIX]{x1D6E5}_{v}}Y_{v}$ and denote by $\overline{\unicode[STIX]{x1D6E5}}_{v}$ the strict transform of $\unicode[STIX]{x1D6E5}_{v}$ in $Bl_{\unicode[STIX]{x1D6E4}_{v}}Y_{v}$ .

Corollary 4.3.

  1. (1) The projective varieties $Bl_{\widetilde{\unicode[STIX]{x1D6E4}}_{v}}Bl_{\unicode[STIX]{x1D6E5}_{v}}Y_{v}$ and $Bl_{\overline{\unicode[STIX]{x1D6E5}}_{v}}Bl_{\unicode[STIX]{x1D6E4}_{v}}Y_{v}$ are smooth and isomorphic over $Y_{v}$ .

  2. (2) There exist finite 2:1 morphisms $\widetilde{\unicode[STIX]{x1D700}}_{v}:Bl_{\unicode[STIX]{x1D6E5}_{v}}Y_{v}\rightarrow \widetilde{K}_{v}$ , $\overline{\unicode[STIX]{x1D700}}_{v}:Bl_{\unicode[STIX]{x1D6E4}_{v}}Y_{v}\rightarrow Bl_{\unicode[STIX]{x1D6FA}_{v}}K_{v}$ , $\widehat{\unicode[STIX]{x1D700}}_{1,v}:Bl_{\widetilde{\unicode[STIX]{x1D6E4}}_{v}}Bl_{\unicode[STIX]{x1D6E5}_{v}}Y_{v}\rightarrow Bl_{\widetilde{\unicode[STIX]{x1D6FA}}_{v}}Bl_{\unicode[STIX]{x1D6F4}_{v}}K_{v}$ and $\widehat{\unicode[STIX]{x1D700}}_{2,v}:Bl_{\overline{\unicode[STIX]{x1D6E5}}_{v}}Bl_{\unicode[STIX]{x1D6E4}_{v}}Y_{v}\rightarrow Bl_{\overline{\unicode[STIX]{x1D6F4}}_{v}}Bl_{\unicode[STIX]{x1D6FA}_{v}}K_{v}$ lifting $\unicode[STIX]{x1D700}_{v}$ . Hence, there exists a commutative diagram

    where the diagonal arrows are blow ups.

Proof. (1) Over the inverse images of the smooth locus of $Y_{v}$ , the existence of the isomorphism is trivial, whereas over the inverse images of small Euclidean neighborhoods of the singular points of $Y_{v}$ it follows from item (2) of Lemma 3.7. Since the global blow up is obtained by gluing local blow ups, $Bl_{\widetilde{\unicode[STIX]{x1D6E4}}_{v}}Bl_{\unicode[STIX]{x1D6E5}_{v}}Y_{v}$ and $Bl_{\overline{\unicode[STIX]{x1D6E5}}_{v}}Bl_{\unicode[STIX]{x1D6E4}_{v}}Y_{v}$ are isomorphic over $Y_{v}$ . The smoothness of $Bl_{\overline{\unicode[STIX]{x1D6E5}}_{v}}Bl_{\unicode[STIX]{x1D6E4}_{v}}Y_{v}$ and $Bl_{\overline{\unicode[STIX]{x1D6E5}}_{v}}Bl_{\unicode[STIX]{x1D6E4}_{v}}Y_{v}$ follows from the smoothness of $Bl_{\overline{\unicode[STIX]{x1D6E5}}}Bl_{\unicode[STIX]{x1D6E4}}W$ and $Bl_{\overline{\unicode[STIX]{x1D6E5}}}Bl_{\unicode[STIX]{x1D6E4}}W$ , which was proven in Corollary 3.7. Hence item (1) holds.

(2) The existence of the liftings of the double cover $\unicode[STIX]{x1D700}_{v}$ over the inverse images of $K_{v}\setminus \unicode[STIX]{x1D6FA}_{v}$ is clear. Over the inverse images of a small Euclidean neighborhood in $K_{v}$ of a point of $\unicode[STIX]{x1D6FA}_{v}$ , the existence of the lift follows from Theorem 2.3(a), Corollaries 3.3 and 3.5 and Lemma 3.7(1). Since the lift of a morphism to bimeromorphic varieties is unique, whenever it exists, it is possible to glue the local liftings and obtain the desired global morphism. ◻

Remark 4.4. In Corollary 4.3 we have shown that $Bl_{\unicode[STIX]{x1D6E5}_{v}}Y_{v}$ is a double cover of $\widetilde{K}_{v}$ and that it is branched over $\widetilde{\unicode[STIX]{x1D6F4}}_{v}$ . Moreover, by Corollary 3.5, the projective variety $Bl_{\unicode[STIX]{x1D6E5}_{v}}Y_{v}$ is normal. Since the Picard group of the IHS manifold $\widetilde{K}_{v}$ is torsion-free, there exists a unique such double cover. It follows that $Bl_{\unicode[STIX]{x1D6E5}_{v}}Y_{v}=\widetilde{Y}_{v}$ and $\widetilde{\unicode[STIX]{x1D700}}_{v}=\widetilde{\unicode[STIX]{x1D700}}_{v}$ .

In order to describe the ramification loci of these double coverings, we need to introduce some further notation. In the following corollary we denote by $\widetilde{\unicode[STIX]{x1D6E5}}_{v}\subset Bl_{\unicode[STIX]{x1D6E5}_{v}}Y_{v}$ the exceptional divisor and by $\widehat{\unicode[STIX]{x1D6E5}}_{v}\subset Bl_{\widetilde{\unicode[STIX]{x1D6E4}}_{v}}Bl_{\unicode[STIX]{x1D6E5}_{v}}Y_{v}=Bl_{\overline{\unicode[STIX]{x1D6E5}}_{v}}Bl_{\unicode[STIX]{x1D6E4}_{v}}Y_{v}$ the strict transform of $\widetilde{\unicode[STIX]{x1D6E5}}_{v}$ or, equivalently, the exceptional divisor of the blow up of $Bl_{\unicode[STIX]{x1D6E4}_{v}}Y_{v}$ along $\overline{\unicode[STIX]{x1D6E5}}_{v}$ .

Corollary 4.5.

  1. (1) The branch loci of $\widetilde{\unicode[STIX]{x1D700}}_{v}$ , $\overline{\unicode[STIX]{x1D700}}_{v}$ and $\widehat{\unicode[STIX]{x1D700}}_{1,v}~(=\widehat{\unicode[STIX]{x1D700}}_{2,v})$ are $\widetilde{\unicode[STIX]{x1D6F4}}_{v}$ , $\overline{\unicode[STIX]{x1D6F4}}_{v}$ and $\widehat{\unicode[STIX]{x1D6F4}}_{v}$ , respectively.

  2. (2) The ramification loci of $\widetilde{\unicode[STIX]{x1D700}}_{v}$ , $\overline{\unicode[STIX]{x1D700}}_{v}$ and $\widehat{\unicode[STIX]{x1D700}}_{1,v}~(=\widehat{\unicode[STIX]{x1D700}}_{2,v})$ are $\widetilde{\unicode[STIX]{x1D6E5}}_{v}$ , $\overline{\unicode[STIX]{x1D6E5}}_{v}$ and $\widehat{\unicode[STIX]{x1D6E5}}_{v}$ , respectively.

Proof. The statements on the branch loci are determined by the analogous statement proved for the local case. Specifically, (1) follows from Corollaries 3.5 and 3.3 and item (1) of Corollary 3.7.

Since the ramification locus of $\unicode[STIX]{x1D700}_{v}$ is $\unicode[STIX]{x1D6E5}_{v}$ and its branch locus is $\unicode[STIX]{x1D6F4}_{v}$ , (2) follows from (1) and from the commutativity of the diagram in item (2) of Corollary 4.3.◻

In the final part of this section on the global geometry of the double covers induced by $\unicode[STIX]{x1D700}_{v}$ , we compare their ramification and their branch loci and discuss the associated involutions.

Remark 4.6. Since $\unicode[STIX]{x1D6F4}_{v}\simeq A\times A^{\vee }/\pm 1$ , $\widetilde{\unicode[STIX]{x1D6F4}}_{v}$ has an $A_{1}$ singularity along its singular locus (see Remark 2.2). Moreover, $\overline{\unicode[STIX]{x1D6F4}}_{v}$ and $\widehat{\unicode[STIX]{x1D6F4}}_{v}$ are smooth (see Proposition 2.6) and the branch loci of $\unicode[STIX]{x1D700}_{v}$ , $\widetilde{\unicode[STIX]{x1D700}}_{v}$ , $\overline{\unicode[STIX]{x1D700}}_{v}$ and $\widehat{\unicode[STIX]{x1D700}}_{1,v}~(=\widehat{\unicode[STIX]{x1D700}}_{2,v})$ are normal. Hence these double covers induce isomorphisms $\unicode[STIX]{x1D6E5}_{v}\simeq \unicode[STIX]{x1D6F4}_{v}$ , $\widetilde{\unicode[STIX]{x1D6E5}}_{v}\simeq \widetilde{\unicode[STIX]{x1D6F4}}_{v}$ , $\overline{\unicode[STIX]{x1D6E5}}_{v}\simeq \overline{\unicode[STIX]{x1D6F4}}_{v}$ and $\widehat{\unicode[STIX]{x1D6E5}}_{v}\simeq \widehat{\unicode[STIX]{x1D6F4}}_{v}$ .

Remark 4.7. Remark 3.4 implies that $Bl_{\unicode[STIX]{x1D6E4}_{v}}Y_{v}$ and $Bl_{\widetilde{\unicode[STIX]{x1D6E4}}_{v}}Bl_{\unicode[STIX]{x1D6E5}_{v}}Y_{v}=Bl_{\overline{\unicode[STIX]{x1D6E5}}_{v}}Bl_{\unicode[STIX]{x1D6E4}_{v}}Y_{v}$ are smooth, Corollary 3.5 implies that $Bl_{\unicode[STIX]{x1D6E5}_{v}}Y_{v}$ is normal and, by Theorem 4.2, $Y_{v}$ is normal too. Hence the finite $2:1$ morphisms $\unicode[STIX]{x1D700}_{v}$ , $\widetilde{\unicode[STIX]{x1D700}}_{v}$ , $\overline{\unicode[STIX]{x1D700}}_{v}$ and $\widehat{\unicode[STIX]{x1D700}}_{1,v}(=\widehat{\unicode[STIX]{x1D700}}_{2,v})$ induce regular involutions $\unicode[STIX]{x1D70F}_{v}$ , $\widetilde{\unicode[STIX]{x1D70F}}_{v}$ , $\overline{\unicode[STIX]{x1D70F}}_{v}$ , $\widehat{\unicode[STIX]{x1D70F}}_{1,v}$ and $\widehat{\unicode[STIX]{x1D70F}}_{2,v}$ on $Y_{v}$ , $Bl_{\unicode[STIX]{x1D6E5}_{v}}Y_{v}$ , $Bl_{\unicode[STIX]{x1D6E4}_{v}}Y_{v}$ , $Bl_{\widetilde{\unicode[STIX]{x1D6E4}}_{v}}Bl_{\unicode[STIX]{x1D6E5}_{v}}Y_{v}$ and $Bl_{\overline{\unicode[STIX]{x1D6E5}}_{v}}Bl_{\unicode[STIX]{x1D6E4}_{v}}Y_{v}$ , respectively. Recall that $K_{v}$ is normal, and hence so is $Bl_{\unicode[STIX]{x1D6FA}_{v}}K_{v}$ by Proposition 2.4. As for $\widetilde{K}_{v}$ and $Bl_{\widetilde{\unicode[STIX]{x1D6FA}}_{v}}Bl_{\unicode[STIX]{x1D6F4}_{v}}K_{v}=Bl_{\overline{\unicode[STIX]{x1D6F4}}_{v}}Bl_{\unicode[STIX]{x1D6FA}_{v}}K_{v}$ , they are both smooth. It follows that the morphisms $\unicode[STIX]{x1D700}_{v}$ , $\widetilde{\unicode[STIX]{x1D700}}_{v}$ , $\overline{\unicode[STIX]{x1D700}}_{v}$ and $\widehat{\unicode[STIX]{x1D700}}_{1,v}(=\widehat{\unicode[STIX]{x1D700}}_{2,v})$ can be identified with the quotient maps of the respective involutions $\unicode[STIX]{x1D70F}_{v}$ , $\widetilde{\unicode[STIX]{x1D70F}}_{v}$ , $\overline{\unicode[STIX]{x1D70F}}_{v}$ , $\widehat{\unicode[STIX]{x1D70F}}_{1,v}$ and $\widehat{\unicode[STIX]{x1D70F}}_{2,v}$ .

5 The birational geometry of $Y_{v}$

In this section we describe the global geometry of $Y_{v}$ , we show that it is birational to an IHS manifold of K3 $^{[n]}$ type, and we describe the birational map explicitly.

In the first part of the section, we consider the special case where $A$ is a principally polarized abelian surface, whose Néron–Severi group is generated by the principal symmetric polarization $\unicode[STIX]{x1D6E9}$ . As is well known, the linear system $|2\unicode[STIX]{x1D6E9}|$ defines a morphism $g_{|2\unicode[STIX]{x1D6E9}|}:A\rightarrow |2\unicode[STIX]{x1D6E9}|^{\vee }\simeq \mathbb{P}^{3}$ whose image is the singular Kummer surface $\text{Kum}_{s}$ of $A$ , a nodal quartic surface isomorphic to the quotient $A/\pm 1$ . The smooth Kummer surface $S$ of $A$ is the blow up of $\text{Kum}_{s}$ along the singular locus $A[2]$ .

We are going to show that in this case $Y_{(0,2\unicode[STIX]{x1D6E9},2)}$ is birational to the Hilbert scheme $S^{[3]}$ , for which we will need the following remark on known properties of the linear system $|2\unicode[STIX]{x1D6E9}|$ .

Remark 5.1. (1) The locus of $|2\unicode[STIX]{x1D6E9}|$ parametrizing singular curves consists of $17$ irreducible divisors: the divisor $R$ parametrizing reducible curves and, for any $2$ -torsion point $\unicode[STIX]{x1D6FC}\in A[2]$ , the divisor $N_{\unicode[STIX]{x1D6FC}}$ parametrizing curves passing through $\unicode[STIX]{x1D6FC}$ . The divisor $R$ is isomorphic to $\text{Kum}_{s}$ and a general point of $R$ corresponds to a curve of the form $\unicode[STIX]{x1D6E9}_{x}\cup \unicode[STIX]{x1D6E9}_{-x}$ , where $\unicode[STIX]{x1D6E9}_{x}$ and $\unicode[STIX]{x1D6E9}_{-x}$ meet transversally outside of $A[2]$ . For every $\unicode[STIX]{x1D6FC}\in A[2]$ , the divisor $N_{\unicode[STIX]{x1D6FC}}$ is isomorphic to $\mathbb{P}^{2}$ and parametrizes curves whose images in $\text{Kum}_{s}$ are plane sections through the singular point $g_{|2\unicode[STIX]{x1D6E9}|}(\unicode[STIX]{x1D6FC})$ . The general point of $N_{\unicode[STIX]{x1D6FC}}$ corresponds to a curve $C$ that is a double cover of a quartic plane curve with precisely one node; this double covers ramifies over the node and therefore $C$ has a node in $\unicode[STIX]{x1D6FC}$ and no other singularity.

(2) For the natural choices in the definition of the map

$$\begin{eqnarray}\mathbf{a}_{(0,2\unicode[STIX]{x1D6E9},2)}:M_{(0,2\unicode[STIX]{x1D6E9},2)}(A,\unicode[STIX]{x1D6E9})\rightarrow A\times A^{\vee }\end{eqnarray}$$

(see Introduction, formula (1.2)), the subvariety $K_{(0,2\unicode[STIX]{x1D6E9},2)}:=\mathbf{a}_{(0,2\unicode[STIX]{x1D6E9},2)}^{-1}(0,0)\subset M_{(0,2\unicode[STIX]{x1D6E9},2)}(A,\unicode[STIX]{x1D6E9})$ parametrizes sheaves whose determinant is equal to ${\mathcal{O}}(2\unicode[STIX]{x1D6E9})$ and whose second Chern class sums up to $0\in A$ . Since $M_{(0,2\unicode[STIX]{x1D6E9},2)}(A,\unicode[STIX]{x1D6E9})$ parametrizes pure dimension- $1$ sheaves, there exists a regular morphism $t:K_{v}\rightarrow |2\unicode[STIX]{x1D6E9}|\simeq \mathbb{P}^{3}$ , called the support morphism, which to every polystable sheaf associates its Fitting subscheme (see [Reference Le PotierLep93]). The morphism $t$ is surjective and, since $K_{(0,2\unicode[STIX]{x1D6E9},2)}$ has a resolution that is an IHS manifold, all its fibers are three-dimensional.

(3) Since the polarization $\unicode[STIX]{x1D6E9}$ is symmetric, $-1^{\ast }$ induces an involution on the moduli space $M_{(0,2\unicode[STIX]{x1D6E9},2)}(A,\unicode[STIX]{x1D6E9})$ whose fixed locus contains the variety $K_{(0,2\unicode[STIX]{x1D6E9},2)}$ . Indeed, any smooth curve $C\in |2\unicode[STIX]{x1D6E9}|$ is an étale double cover of its image $g_{|2\unicode[STIX]{x1D6E9}|}(C)$ . The pull back to $C$ of any degree- $3$ line bundle on $g_{|2\unicode[STIX]{x1D6E9}|}(C)$ is a stable sheaf of $K_{(0,2\unicode[STIX]{x1D6E9},2)}$ which is $-1^{\ast }$ -invariant. Moreover, the pull backs of two line bundles on $g_{|2\unicode[STIX]{x1D6E9}|}(C)$ are isomorphic if and only if the two line bundles differ by a $2$ -torsion line bundle defining the étale double cover $C\rightarrow g_{|2\unicode[STIX]{x1D6E9}|}(C)$ . Hence, there exists a six-dimensional algebraic subset of $K_{(0,2\unicode[STIX]{x1D6E9},2)}$ that is fixed by the involution and hence, by closure of the fixed locus, the whole $K_{(0,2\unicode[STIX]{x1D6E9},2)}$ is fixed.

(4) If $C\in |2\unicode[STIX]{x1D6E9}|$ is smooth or general in $R$ or $N_{\unicode[STIX]{x1D6FC}}$ , the general point of $t^{-1}(C)$ represents a sheaf that is locally free on its support. This holds because, for any nodal curve $C$ , any torsion-free sheaf on $C$ that is not locally free is the limit of locally free sheaves on $C$ varying in a family parametrized by $\mathbb{P}^{1}$ . Since any $\mathbb{P}^{1}$ has to be contracted by $\mathbf{a}_{(0,2\unicode[STIX]{x1D6E9},2)}$ , it follows that it has to be contained in $K_{(0,2\unicode[STIX]{x1D6E9},2)}$ and the claim follows.

(5) The inverse image on $K_{(0,2\unicode[STIX]{x1D6E9},2)}$ of an irreducible surface contained in $|2\unicode[STIX]{x1D6E9}|\simeq \mathbb{P}^{3}$ is irreducible. Since $t$ is equidimensional, it suffices to prove that $t^{-1}(C)$ is irreducible for any curve $C$ that is smooth or general in $R$ or in $N_{\unicode[STIX]{x1D6FC}}$ . If $C$ is such a curve, the locus