Hilbert squares of \(K3\) surfaces and Debarre-Voisin varieties (Q2184860)

Let \(V\) be a \(10\)-dimensional complex vector space. From a trivector \(\sigma\in\bigwedge^{3}V^{\vee}\), we associate a canonically polarized subvariety \(K_{\sigma}\) of \(\mathrm{Gr}(6,V)\), called the Debarre-Voisin variety. For general \(\sigma\), \(K_{\sigma}\) form a local complete family of hyperkähler of \(K3^{[2]}\)-type. The GIT quotient \(\mathscr{M}_{DV}:=\mathbb{P}(\bigwedge^{3}V^{\vee})//\mathrm{SL}(\bigwedge^{3}V^{\vee})\) is the coarse moduli space of trivectors. By composition with the period map, we can construct a rational map \(q:\mathscr{M}_{DV}\dashrightarrow\mathcal{F}\), where \(\mathcal{F}\) is the period domain of polarized varieties of type \((K_{\sigma},\mathcal{O}_{K_{\sigma}}(1))\). There exists a resolution of \(q\), \(\overline{q}:\widetilde{\mathscr{M}_{DV}}\twoheadrightarrow\overline{\mathcal{F}}\), where \(\overline{\mathcal{F}}\) is the Baily-Borel projective compactification of \(\mathcal{F}\) and \(\widetilde{\mathscr{M}_{DV}}\) is obtained from the blow up of \(\mathbb{P}(\bigwedge^{3}V^{\vee})\) along trivectors with positive-dimensional stabilizers. An HLS divisor is an irreducible hypersurface of \(\overline{\mathcal{F}}\) which is the image of an exceptional divisor of \(\widetilde{\mathscr{M}_{DV}}\rightarrow\mathscr{M}_{DV}\). The main result of the article under review is that the Heegner divisors \(\mathscr{D}_{2},\mathscr{D}_{6},\mathscr{D}_{10},\mathscr{D}_{18}\) are HLS divisors. These divisors are studied one by one. In the case \(\mathscr{D}_{6}\), the strategy is the following. Define a trivector \(\sigma\) whose Debarre-Voisin variety \(K_{\sigma}\) is smooth but has dimension \(6\) (this was proved by \textit{P. Hivert} in [Ann. Inst. Fourier 61, No. 5, 2121--2138 (2011; Zbl 1298.14050)]). A general \(1\)-parameter deformation of \(\sigma\) defines a family of Debarre-Voisin varieties with central fiber \(S^{[2]}\), where \(S\) is a surface \(K3\) of degree \(6\). The subvariety \(S^{[2]}\subset K_{\sigma}\) is the zero locus of a rank \(2\) excess bundle described explicitly. In the case \(\mathscr{D}_{18}\) the authors apply a very similar strategy. In the case \(\mathscr{D}_{10}\), the authors start from a degree \(10\) general polarized \(K3\) surface \((S,L)\). Such a variety is the anticanonical section of a Fano threefold \(X\subset\mathrm{Gr}(2,\mathbb{C}^{5})\) of index \(2\). Moreover, they construct a vector bundle on \(X^{[2]}\) such that the global sections define a birational map from \(X^{[2]}\) to a certain smooth subspace \(K_{1}\subset\mathrm{Gr}(6,V)\). The restriction of such a map to \(S^{[2]}\) is regular outside a smooth surface. After that, they define a trivector \(\sigma\) whose Debarre-Voisin variety \(K_{\sigma}\) has \(K_{1}\) as component. A general \(1\)-parameter family of \(\sigma\) gives, after a pull back by a finite cover, a family of Debarre-Voisin varieties where the central fiber is the polarized variety \((S^{[2]},2L-3\delta)\). In the final case, they construct a trivector \(\sigma\) whose Debarre-Voisin variety is neither reduced nor irreducible. A general \(1\)-parameter family of \(\sigma\) gives, after a pull back by a finite cover, a family of Debarre-Voisin varieties where the central fiber is the polarized variety \((\mathcal{M}_{S}(0,L,1),6L-5\delta)\). That is, the moduli space of \(L\)-semistable pure sheaves of Mukai vector \((0,L,1)\) on a \(K3\) surface \((S,L)\) of degree \(2\). In the last section, some partial result of the case \(\mathscr{D}_{30}\) are described.