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.◻

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 $\widehat{\unicode[STIX]{x1D6FA}}\simeq \mathbb{P}(\operatorname{Sym}_{G}^{2}{\mathcal{U}})$ .◻

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.◻

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

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.◻

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

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

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).◻

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. ◻

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].◻

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.◻

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. ◻

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.◻

Proof. Let $D$ be the pull back on $S$ of a plane section of $\text{Kum}_{s}$ . We are going to show that $\widetilde{Y}_{(0,\unicode[STIX]{x1D6E9},2)}$ is birational to the smooth projective moduli space $M_{(0,D,1)}$ parametrizing sheaves on $S$ with Mukai vector $(0,D,1)$ and which are stable with respect to a fixed $(0,D,1)$ -generic polarization. The moduli space $M_{(0,D,1)}$ is well known to be birational to $S^{[3]}$ (see [Reference BeauvilleBea99, Proposition 1.3]).

By construction, there exists an isomorphism between linear systems $\unicode[STIX]{x1D713}:|D|\rightarrow |2\unicode[STIX]{x1D6E9}|$ . Moreover, any sheaf $F\in M_{(0,D,1)}$ , whose support is a smooth curve, may be seen as a sheaf on $\text{Kum}_{s}$ and its pull back to $A$ is a stable sheaf of $K_{(0,2\unicode[STIX]{x1D6E9},2)}$ . It follows that there exists a commutative diagram

(5.1)

where $s$ and $t$ are the two support morphisms. If $C\in |2\unicode[STIX]{x1D6E9}|$ is a smooth curve, it is a connected étale double cover of the smooth curve $g_{|2\unicode[STIX]{x1D6E9}|}(C)$ , and since $g_{|2\unicode[STIX]{x1D6E9}|}(C)\,\cap \,A[2]=\emptyset$ , it can be considered as a curve in $|D|$ . The restriction of $\unicode[STIX]{x1D711}$ on $s^{-1}(g_{|2\unicode[STIX]{x1D6E9}|}(C))\simeq \text{Pic}^{3}(g_{|2\unicode[STIX]{x1D6E9}|}(C))$ is therefore well defined and gives an étale double cover of $(t\circ f_{(0,2\unicode[STIX]{x1D6E9},2)})^{-1}(C)\simeq t^{-1}(C)\subset \text{Pic}^{6}(C)$ (see (3) of Remark 5.1). This shows that $\unicode[STIX]{x1D711}$ is a rational map of degree $2$ .

In order to compare $\widetilde{Y}_{(0,2\unicode[STIX]{x1D6E9},2)}$ and $M_{(0,D,1)}$ , we need to determine the branch divisor $B$ of $\unicode[STIX]{x1D711}$ , i.e. the divisor on $\widetilde{K}_{v}$ where a resolution of the indeterminacy of $\unicode[STIX]{x1D711}$ is not étale. We have already seen that $B$ has to parametrize sheaves supported on singular curves.

Let $U\subset M_{(0,D,1)}$ be the biggest open subset where $\unicode[STIX]{x1D711}$ extends to a regular morphism. As $M_{(0,D,1)}$ and $K_{(0,2\unicode[STIX]{x1D6E9},2)}$ have trivial canonical bundle, the differential of $\unicode[STIX]{x1D711}$ is an isomorphism at any point of $U$ . As a consequence, $\unicode[STIX]{x1D711}$ does not contract any positive dimensional subvariety and $\unicode[STIX]{x1D711}(U)\subset \widetilde{K}_{(0,2\unicode[STIX]{x1D6E9},2)}$ is an open subset.

We claim that the open subset $\unicode[STIX]{x1D711}(U)$ intersects any divisor of $\widetilde{K}_{(0,2\unicode[STIX]{x1D6E9},2)}$ , with the possible exception of $\widetilde{\unicode[STIX]{x1D6F4}}_{(0,2\unicode[STIX]{x1D6E9},2)}$ . Since we have already shown that $(t\circ f_{(0,2\unicode[STIX]{x1D6E9},2)})^{-1}(C)\in \unicode[STIX]{x1D711}(U)$ if $C$ is smooth, it remains to check this statement for divisors contained in $(t\,\circ \,f_{(0,2\unicode[STIX]{x1D6E9},2)})^{-1}(R)$ and $(t\,\circ \,f_{(0,2\unicode[STIX]{x1D6E9},2)})^{-1}(N_{\unicode[STIX]{x1D6FC}})$ . As $\unicode[STIX]{x1D6F4}_{(0,2\unicode[STIX]{x1D6E9},2)}\subset t^{-1}(R)$ , by (5) of Remark 5.1, the divisor $(t\,\circ \,f_{(0,2\unicode[STIX]{x1D6E9},2)})^{-1}(R)$ is the union of $\widetilde{\unicode[STIX]{x1D6F4}}_{(0,2\unicode[STIX]{x1D6E9},2)}$ and the strict transform of $t^{-1}(R)$ . Finally, $(t\,\circ \,f_{(0,2\unicode[STIX]{x1D6E9},2)})^{-1}N_{\unicode[STIX]{x1D6FC}}$ is irreducible.

The general point $F$ of $t^{-1}(R)$ represents a line bundle supported on the general curve $C$ of  $R$ . As $C$ does not intersect $A[2]$ , its image $g_{|2\unicode[STIX]{x1D6E9}|}(C)\simeq C/\pm 1$ may be seen as a curve in $D$ and, as in the smooth case, $F$ descends to a line bundle on $g_{|2\unicode[STIX]{x1D6E9}|}(C)$ . Hence $\unicode[STIX]{x1D711}(U)$ intersects the strict transform of $t^{-1}(R)$ .

Finally, by the commutativity of diagram (5.1), $\unicode[STIX]{x1D711}$ sends an open subset of $(\unicode[STIX]{x1D713}\circ s)^{-1}(N_{\unicode[STIX]{x1D6FC}})$ to the irreducible divisor $(t\circ f_{(0,2\unicode[STIX]{x1D6E9},2)})^{-1}N_{\unicode[STIX]{x1D6FC}}$ . Since $(\unicode[STIX]{x1D713}\circ s)^{-1}(N_{\unicode[STIX]{x1D6FC}})$ cannot be contracted, $\unicode[STIX]{x1D711}(U)$ also intersects $(t\circ f_{(0,2\unicode[STIX]{x1D6E9},2)})^{-1}N_{\unicode[STIX]{x1D6FC}}$ . This completes the proof of our claim.

Let $r:N\rightarrow \widetilde{K}_{v}$ be a resolution of the indeterminacy of $\unicode[STIX]{x1D711}$ , hence $N$ is a smooth projective variety such that there exists a commutative diagram

where $b$ is birational and induces an isomorphism between $b^{-1}(U)$ and $U$ . Let $U^{\prime }\subset \widetilde{K}_{(0,2\unicode[STIX]{x1D6E9},2)}\setminus \widetilde{\unicode[STIX]{x1D6FA}}_{v}$ be the open subset where the fibers of $\unicode[STIX]{x1D709}$ are zero-dimensional. Note that, since $N$ and $\widetilde{K}_{(0,2\unicode[STIX]{x1D6E9},2)}$ are smooth, $r$ is flat over $U^{\prime }$ . Therefore $r^{-1}(U^{\prime })$ is a flat ramified double cover of $U^{\prime }$ and, since $\widetilde{K}_{(0,2\unicode[STIX]{x1D6E9},2)}\setminus U^{\prime }$ has codimension at least $2$ and $\text{Pic}(\widetilde{K}_{(0,2\unicode[STIX]{x1D6E9},2)})$ is torsion-free, this double cover is determined by its branch divisor $B$ , i.e. by the locus where fibers of $r$ are length- $2$ non-reduced subschemes.

We already know that if $p\in U^{\prime }\,\cap \,\unicode[STIX]{x1D711}(U)$ the fiber $r^{-1}(p)$ has at least one component consisting of a reduced point of $U$ . Hence $B$ is a divisor contained in $U^{\prime }\setminus \unicode[STIX]{x1D711}(U)$ : therefore, by our claim, either $B$ is empty or $B=U^{\prime }\cap \widetilde{\unicode[STIX]{x1D6F4}}_{(0,2\unicode[STIX]{x1D6E9},2)}$ . The first case is impossible becouse $U^{\prime }$ is simply connected and $N$ is irreducible. In the second case $r^{-1}(U^{\prime })$ is the unique double cover of $U^{\prime }$ ramified over $U^{\prime }\cap \widetilde{\unicode[STIX]{x1D6F4}}_{(0,2\unicode[STIX]{x1D6E9},2)}$ , hence $r^{-1}(U^{\prime })$ is isomorphic to $\unicode[STIX]{x1D700}_{v}^{-1}(U^{\prime })\subset \widetilde{Y}_{(0,2\unicode[STIX]{x1D6E9},2)}$ .◻

Proof. Note that the ramification locus of $\unicode[STIX]{x1D700}_{v}:Y_{v}\rightarrow K_{v}$ has codimension $2$ . It follows that the canonical divisor of $Y_{v}$ is trivial and the canonical divisor of $Bl_{\unicode[STIX]{x1D6E4}_{v}}Y_{v}$ is supported on $\overline{\unicode[STIX]{x1D6E4}}_{v}$ . As the normal bundle of $I_{i,v}$ in $Bl_{\unicode[STIX]{x1D6E4}_{v}}Y_{v}$ has degree $-1$ on the fibers of both the $\mathbb{P}^{2}$ fibrations of $I_{i,v}$ , by adjunction, the canonical bundle of the smooth variety $Bl_{\unicode[STIX]{x1D6E4}_{v}}Y_{v}$ is $2\sum _{i=1}^{256}I_{i,v}$ .

Let $r_{i}$ be a line contained in a fiber of $p_{i}$ and let $l_{i}$ be a line contained in a fiber of the other $\mathbb{P}^{2}$ fibration of $I_{i,v}$ . A priori, it is not clear whether $r_{i}$ and $l_{i}$ are numerically equivalent. Nevertheless, since $Y_{v}$ is projective and $r_{i}$ and $l_{i}$ generate the cone of effective curves on $I_{i,v}$ , the set $\{r_{i}\}$ represents $256$ $K_{Bl_{\unicode[STIX]{x1D6E4}_{v}}Y_{v}}$ -negative extremal rays of the Mori cone of $Bl_{\unicode[STIX]{x1D6E4}_{v}}Y_{v}$ . If $r_{i}$ and $l_{i}$ are equivalent, the contraction of $r_{i}$ contracts $I_{i,v}$ to a point admitting a Zariski neighborhood isomorphic to a Zariski neighborhood of the $i$ th singular point of $\unicode[STIX]{x1D6E4}_{v}$ in the normal variety $Y_{v}$ . If $r_{i}$ and $l_{i}$ are independent, the contraction of $r_{i}$ can be identified with the Nakano contraction restricting to $p_{i}$ on $I_{i,v}$ .

In any case, the contraction of $r_{i}$ is divisorial and, by [Reference Kollar and MoriKM98, Corollary 3.18], it produces only $\mathbb{Q}$ -factorial singularities. Hence, after $256$ extremal contractions we terminate with a $\mathbb{Q}$ -factorial variety with trivial canonical divisor and with terminal singularities. If $v=(0,2\unicode[STIX]{x1D6E9},2)$ , by Lemma 5.2, the variety $M_{(0,2\unicode[STIX]{x1D6E9},2)}$ is a minimal model of the IHS manifold $S^{[3]}$ and by a theorem due to Greb, Lehn and Rollenske (see [Reference Greb, Lehn and RollenskeGLR13, Proposition 6.4]) it is an IHS manifold. In particular $r_{i}$ and $l_{i}$ are always numerically independent and this IHS manifold is isomorphic to the Nakano contraction $\text{}\underline{Y}_{v}$ .

To deal with the general case, recall from [Reference Perego and RapagnettaPR13, Theorem 1.6] (and its proof) that the singular variety $K_{v}$ can be deformed to $K_{(0,2\unicode[STIX]{x1D6E9},2)}$ using only isomorphisms induced by Fourier–Mukai transform and locally trivial deformations induced by deformation of the underlying abelian surface (see [Reference Perego and RapagnettaPR13, Proposition 2.16]).

Extending the construction of Theorem 4.2 to the case of a locally trivial deformation, it is also possible to deform $Y_{v}$ to $Y_{(0,2\unicode[STIX]{x1D6E9},2)}$ by a locally trivial deformation. By blowing up the subvariety consisting of singular points of all fibers in the total space of the deformation, we get that $Bl_{\unicode[STIX]{x1D6E4}_{v}}Y_{v}$ can be deformed to $Bl_{\unicode[STIX]{x1D6E4}_{(0,2\unicode[STIX]{x1D6E9},2)}}Y_{(0,2\unicode[STIX]{x1D6E9},2)}$ . Up to an étale base change on the base of the deformation, we may also assume that the exceptional divisor consists of $256$ connected components ${\mathcal{I}}_{i}$ , each of which has two fibrations and one of them restricts to $p_{i}$ on $I_{i,v}$ . Applying again Nakano’s theorem, the ${\mathcal{I}}_{i}$ may be contracted respecting the chosen fibration.

As a consequence, the complex manifold $\text{}\underline{Y}_{v}$ obtained from $Bl_{\unicode[STIX]{x1D6E4}_{v}}Y_{v}$ by contracting the $r_{i}$ is deformation equivalent (via smooth deformations) to an IHS manifold $\text{}\underline{Y}_{(0,2\unicode[STIX]{x1D6E9},2)}$ that is birational to $S^{[3]}$ . It remains to show that $\text{}\underline{Y}_{v}$ is projective. As in the case $v=(0,2\unicode[STIX]{x1D6E9},2)$ , it suffices to show that $r_{i}$ and $l_{i}$ are numerically independent. This is true because parallel transport preserves numerical independence and the analogous statement has been shown to hold on $Bl_{\unicode[STIX]{x1D6E4}_{(0,2\unicode[STIX]{x1D6E9},2)}}Y_{(0,2\unicode[STIX]{x1D6E9},2)}$ .◻

Proof. By Proposition 5.3, $\text{}\underline{Y}_{v}$ is deformation equivalent to the Hilbert scheme parametrizing length- $3$ subschemes on a K3 surface, hence its odd Betti numbers are zero. By construction, $\overline{Y}_{v}$ is the blow up of $\text{}\underline{Y}_{v}$ along $256$ disjoint projective spaces. Since the odd cohomology of the projective space is trivial, the same holds for $\overline{Y}_{v}$ . By definition, $\widehat{Y}_{v}$ is the blow up of $\overline{Y}_{v}$ along $\overline{\unicode[STIX]{x1D6E5}}_{v}$ . We have already recalled that the ramification locus $\overline{\unicode[STIX]{x1D6E5}}_{v}$ of $\overline{\unicode[STIX]{x1D700}}_{v}$ is isomorphic to the corresponding branch locus $\overline{\unicode[STIX]{x1D6F4}}_{v}$ , which, by Remark 4.6, is isomorphic to $(Bl_{(A\times A^{\vee })[2]}(A\times A^{\vee }))/\pm 1$ . As the odd cohomology classes of a torus are always antiinvariant under the action of $\pm 1$ , the odd Betti numbers of $\widehat{Y}_{v}$ are zero. Since $\unicode[STIX]{x1D70C}_{v}\circ \widehat{\unicode[STIX]{x1D700}}_{v}:\widehat{Y}_{v}\rightarrow \widetilde{K}_{v}$ is a regular surjective map between smooth projective varieties, the rational cohomology of $\widetilde{K}_{v}$ injects into the rational cohomology of $\widehat{Y}_{v}$ . Hence the odd Betti numbers of $\widetilde{K}_{v}$ are zero.

We already know that $h^{2}(\widetilde{K}_{v})=8$  [Reference O’GradyO’Gr03] and that $\unicode[STIX]{x1D712}_{\text{top}}(\widetilde{K}_{v})=1920$  [Reference RapagnettaRap04]. The result follows using Salamon’s formula [Reference SalamonSal96], which gives a linear relation among the Betti numbers of a $2n$ -dimensional irreducible holomorphic symplectic manifold