1 Introduction↩︎

Hyperkähler manifolds, stemming from the Beauville–Bogomolov decomposition theorem as one of the building blocks for manifolds with trivial canonical bundle [1], have attracted considerable attention in the last decades. In the context of projective hyperkähler manifolds, one of the most challenging problems is to explicitly describe locally complete families. It is thus of compelling interest to understand linear systems on hyperkähler manifolds, their positivity properties (such as very ampleness and behaviors of higher syzygies), and the geometry of the corresponding projective models.

In the 2-dimensional case, where hyperkähler manifolds are nothing but \(K3\) surfaces, Mukai described in a series of influential works [2]–[6] projective models for general polarized \(K3\) surfaces of low degree. On the other hand, Saint-Donat’s classical results [7] characterize very ampleness numerically, and describe the equations defining \(K3\) surfaces in arbitrary degree. Furthermore, Voisin’s groundbreaking proof of generic Green’s conjecture for canonical curves [8], [9] determines the entire Betti diagram of a \(K3\) surface of Picard rank one (embedded in projective space via the primitive linear system).

For higher-dimensional hyperkähler manifolds, only a few explicit descriptions of locally complete families are known [10]–[15] (and [16], [17] as well). Darkness also predominates in the study of equations and syzygies, in contrast to the case of abelian varieties, where, starting with the pioneering work of Mumford [18], remarkable results (e.g. [19]–[22]) have been obtained in the last 60 years. Nevertheless, with the only evidence of \(K3\) surfaces, many of these results are expected to admit parallels in the hyperkähler setting and thus constitute a source of inspiration (see [23] as an instance of this line of thought for manifolds of generalized Kummer type).

In the sequel, we will restrict ourselves to hyperkähler fourfolds of \(K3^{[2]}\)-type (namely, those that are deformation of the Hilbert square of a \(K3\) surface). Following [24], we denote by \(\mathcal{M}_{2d}^{(\gamma)}\) the moduli space of polarized fourfolds \((X, H)\) of \(K3^{[2]}\)-type having square \(q(H)=2d\) and divisibility \(\mathrm{div}(H)=\gamma\) with respect to the Beauville–Bogomolov–Fujiki quadratic form \(q\) on \(H^2(X,\mathbb{Z})\).¹ In the absence of information for a general element in \(\mathcal{M}_{2d}^{(\gamma)}\), it is natural to specialize to actual Hilbert squares of \(K3\) surfaces equipped with appropriate geometric polarizations, and study their positivity properties that might be preserved under deformation.

More precisely, as proven by Beauville [1], for any \(K3\) surface \(S\) there is an isomorphism \[\mathop{\mathrm{Pic}}(S^{[2]})\cong\mathop{\mathrm{Pic}}(S)\oplus\mathbb{Z}\cdot \delta,\] where \(2\delta\) is the class of the divisor in \(S^{[2]}\) parametrizing non-reduced subschemes. If we denote by \(L_2\in\mathop{\mathrm{Pic}}(S^{[2]})\) the line bundle induced by \(L\in\mathop{\mathrm{Pic}}(S)\), then linear systems of the form \(|L_2-\delta|\)—sending a pair of points in \(S\) to the line they span in \(\mathbb{P}(H^0(L)^\vee)\)—give a natural way of testing basepoint-freeness and very ampleness. For example, it can be shown [24] that a general member of \(\mathcal{M}_{2d}^{(\gamma)}\) with square \(2d\ge 6\) (and either divisibility \(1\) or \(2\)) is very ample. On the other hand, the polarization is never very ample in the case of square \(2\) [12], [25], so this leaves polarizations of square \(4\) (that is, general members of \(\mathcal{M}_4^{(1)}\)) as the only remaining case where the very ampleness is not decided (see also [26]).

The goal of the present paper is to analyze complete linear systems of the form \(|L_2-2\delta|\). Given a very ample line bundle \(L\) on a \(K3\) surface \(S\), \(|L_2-2\delta|\) can be identified with the space of quadric hypersurfaces in \(\mathbb{P}(H^0(L)^\vee)\) containing \(S\), and the induced map \[\label{intro-linearsystem} S^{[2]}\dasharrow \mathbb{P}\left(H^0(S^{[2]},L_2-2\delta)^\vee\right)\tag{1}\] sends a length-2 subscheme \(\xi\) to the hyperplane of quadrics containing the line \(\langle\xi\rangle\) that it spans.

By specializing to \(K3\) surfaces of Picard rank one (in which case \(L_2-2\delta\) is basepoint-free), we will show that the linear systems \(|L_2-2\delta|\) display a rich geometry, and serve as a good testing ground for positivity properties on polarized hyperkähler fourfolds. This fits well in the philosophy that, for a fixed square \(2d\) and divisibility \(1\), the linear systems \(L_2-r\delta\) (in genus \(g=d+r^2+1\)) gain positivity when \(r\) increases. Of course, the limitation with this approach is that these linear systems involve very difficult geometry of the embedding \(S\subset \mathbb{P}(H^0(L)^\vee)\), and so (even in the presence of a Mukai model) become unmanageable very quickly.

Our first contribution, concerning the geometry of \(L_2-2\delta\) in arbitrary genera, is the following.

As an immediate consequence of 1.[thm:L-2delta-very-ample-1] for \(g=7\), we can settle very ampleness for the last remaining case for general hyperkähler fourfolds of \(K3^{[2]}\)-type.

On the other hand, Le Potier’s strange duality conjecture is a statement about duality of sections of determinant line bundles on moduli spaces of sheaves. It has been solved in a wide range of cases (where both moduli spaces parametrize sheaves of rank \(\geq2\)) for generic \(K3\) surfaces, see [27]. In the case \(6\leq g\leq 8\), 1.[thm:L-2delta-very-ample-2] was established by O’Grady [28] through a good understanding of the quadrics containing the Mukai model of \(S\). In our case, well known results of Green [29] on the rank-4 quadrics containing a canonical curve enable us to generalize the argument to all \(g\geq6\).

Next we study the projective geometry of \(S^{[2]}\) under the linear system \(|L_2-2\delta|\) in the cases \(g=7\) and \(g=8\), by making explicit use of their Mukai models. We observe that these two cases share a strikingly similar picture, where the Hilbert square can be realized as a corank-\(2\) degeneracy locus on an ambient homogeneous space of dimension 8. With the help of the Gulliksen–Negård complex [30] resolving the ideal of \(S^{[2]}\) in the ambient homogeneous space, this construction allows us to give a concrete description of the homogeneous ideal and the syzygies of the Hilbert square in the projective space.

Recall from [2] that a general \(K3\) surface \(S\) of genus \(7\) is a \(\mathbb{P}^7\)-linear section of an orthogonal Grassmannian \(\mathop{\mathrm{OGr}}(5, V_{10})\subset\mathbb{P}^{15}\). In particular, there is a natural isomorphism \(V_{10}\cong H^0(S^{[2]},L_2-2\delta)\) for the quadrics in \(\mathbb{P}^7\) containing \(S\). Note also that, by construction, \(\mathbb{P}(V_{10})\) is canonically equipped with a smooth quadric \(Q\) yielding an isomorphism \(\mathbb{P}(V_{10})\cong \mathbb{P}(V_{10}^\vee)\). In this way, the map in 1 gets identified with a closed immersion \[S^{[2]}\mathchoice{\lhook\joinrel\longrightarrow}{\hookrightarrow}{\hookrightarrow}{\hookrightarrow}\mathbb{P}(V_{10})=\mathbb{P}^9\] which turns out to factor through the quadric \(Q\).

Similarly, recall from [2] that a general \(K3\) surface \(S\) of genus \(8\) is a \(\mathbb{P}^8\)-linear section of the Grassmannian \(\mathop{\mathrm{Gr}}(2,V_6)\subset \mathbb{P}({\mathchoice {\textstyle{\bigwedge\mkern-4.5mu^{2}\mkern 1mu}} {\textstyle{\bigwedge\mkern-4.5mu^{2}\mkern 1mu}} {\scriptstyle{\bigwedge\mkern-5mu^{2}}} {\scriptscriptstyle{\bigwedge\mkern-5mu^{2}}}} V_6)\). In this case, quadrics in \(\mathbb{P}^8\) containing \(S\) admit a natural identification \({\mathchoice {\textstyle{\bigwedge\mkern-4.5mu^{2}\mkern 1mu}} {\textstyle{\bigwedge\mkern-4.5mu^{2}\mkern 1mu}} {\scriptstyle{\bigwedge\mkern-5mu^{2}}} {\scriptscriptstyle{\bigwedge\mkern-5mu^{2}}}} V_{6}\cong H^0(S^{[2]},L_2-2\delta)\), so that 1 reads as a closed immersion \[S^{[2]}\mathchoice{\lhook\joinrel\longrightarrow}{\hookrightarrow}{\hookrightarrow}{\hookrightarrow}\mathbb{P}({\mathchoice {\textstyle{\bigwedge\mkern-4.5mu^{2}\mkern 1mu}} {\textstyle{\bigwedge\mkern-4.5mu^{2}\mkern 1mu}} {\scriptstyle{\bigwedge\mkern-5mu^{2}}} {\scriptscriptstyle{\bigwedge\mkern-5mu^{2}}}} V_6^\vee)=\mathbb{P}^{14}\] which turns out to factor through the Grassmannian \(\mathop{\mathrm{Gr}}(2,V_6^\vee)\).

Upon completion of this manuscript, we were informed that 2.[intro-thmC-1] and 3.[intro-thmD-1] were discussed in Benedetti’s doctoral dissertation [31].

After 2 and 3, one is then naturally led to consider the analogous questions on the homogeneous ideal and syzygies for a general deformation of the pair \((S^{[2]},L_2-2\delta)\). In the case of genus \(7\), after performing some deformation-theoretic arguments, we prove that the projective model of a general deformation of \((S^{[2]},L_2-2\delta)\) is not contained in a quadric hypersurface. This allows us to deduce its full minimal graded resolution. Our main results about general members of \(\mathcal{M}_4^{(1)}\) and \(\mathcal{M}_6^{(1)}\), a first step towards the understanding of their projective geometry, read as follows:

We would like to point out that it can be proved that a general fourfold in \(\mathcal{M}_4^{(1)}\) (respectively in \(\mathcal{M}_6^{(1)}\)) is the scheme-theoretical intersection of the \(10\) cubics (respectively of the \(15\) quadrics) alone,² which are the polars of a unique hypersurface of degree \(4\) (respectively of degree \(3\)). Manifestly, these hypersurfaces bear a striking resemblance to the classical Coble hypersurfaces for abelian varieties [32]. A detailed investigation of these Coble type hypersurfaces will appear in our upcoming work [33].

Let us also note that this is in remarkable contrast with the varieties of lines of cubic fourfolds (general elements of \(\mathcal{M}_6^{(2)}\)), where the projective normality can be checked via the Koszul complex; in this case, the 15 quadrics are nothing but the Plücker quadrics containing the Grassmannian \(\mathop{\mathrm{Gr}}(2,6)\), and so are far from cutting out the hyperkähler fourfold.

Finally, it is worth noting that the minimal resolution of the coordinate ring of a general \((X,H)\in \mathcal{M}_4^{(1)}\) behaves exactly as the Hilbert function predicts, in the sense that there are no unexpected syzygies. This behavior suggests that the syzygies for a general polarized hyperkähler fourfold of \(K3^{[2]}\)-type with divisibility \(1\) might follow their surface counterparts and be “as simple as possible”. This leads to some concrete conjectural descriptions, which shall be addressed in a subsequent work. They are included at the end of this paper, together with some other natural observations and remarks arising from our results.

Structure of the paper↩︎

We recall some preliminary results in 2, and notably in 2.4 the description of the locus of quadrics of low rank containing a \(K3\) surface. The description is crucial for the proof of 1, which will be presented in 3. Then we study the two cases of Hilbert squares of \(K3\) surfaces in genus \(7\) and \(8\), as well as their general deformations, in 4 and 5, respectively. Finally, in 6 we raise several questions concerning the geometry of \((S^{[2]},L_2-2\delta)\) in arbitrary genus, as well as the syzygies of their general deformations.

Acknowledgments↩︎

We are grateful to Chiara Camere, Martí Lahoz, Federico Moretti, and Benedetta Piroddi for useful conversations. We also thank Grzegorz and Michał Kapustka for informing us about the results contained in [31], and Vladimiro Benedetti for follow-up discussions.

Despite all proofs being done in the “old-fashioned way”, the usage of free and open-source computer algebra systems Macaulay2 [34], Singular [35], and Sage [36] is instrumental in this project. We would like to thank the developers for their efforts.

Ríos Ortiz was supported by the European Research Council (ERC) under the European Union’s Horizon 2020 research and innovation programme (ERC-2020-SyG-854361-HyperK). Rojas was supported by a Beatriu de Pinós postdoctoral fellowship (no. 2023 BP 00054), the Spanish MICINN project PID2019-104047GB-I00, and the ERC Advanced Grant SYZYGY. Song was supported by a BMS Dirichlet postdoctoral fellowship and the PRIN2022 research grant 2022PEKYBJ.

2 Preliminaries↩︎

2.1 Generalities on hyperkähler manifolds↩︎

We start by recalling some basic properties on hyperkähler manifolds. An excellent introduction to the subject can be found in [24].

A simply connected compact Kähler manifold \(X\) is called a hyperkähler manifold if \(H^0(X,\Omega_X^2)\) is generated by a nowhere degenerate holomorphic \(2\)-form, in other words, a symplectic form. This means the dimension of \(X\) is necessarily even. Let \(X\) be a hyperkähler manifold of dimension \(2m\). There exists a canonical integral primitive quadratic form \(q\mathrel{\vcenter{:}}= q_X\) on \(H^2(X,\mathbb{Z})\) called the Beauville–Bogomolov–Fujiki form of \(X\).

Let \((X,H)\) be a polarized hyperkähler manifold, where \(H\) is an ample class on \(X\). We have the following two invariants of \(H\):

• The square \(q(H)\) with respect to the quadratic form;

• The divisibility \(\mathrm{div}(H)\), which is the generator of the subgroup \(q(H, H^2(X, \mathbb{Z}))\) of \(\mathbb{Z}\).

The theory for moduli spaces of polarized hyperkähler manifolds is very well understood via the global Torelli theorem, thanks to the works of Verbitsky [37] and Markman [38].

By [39] there exists a polynomial \(\mathrm{RR_X}(q)\in\mathbb{Q}[q]\) called the Riemann–Roch polynomial of \(X\), such that for any line bundle \(L\in \mathop{\mathrm{Pic}}(X)\) we have \[\chi(X,L) =\mathrm{RR}_X(q(L)).\] If \(L\) is moreover ample, then thanks to the Kodaira vanishing theorem, the Riemann–Roch polynomial computes the number of global sections \(h^0(X,L)\).

A hyperkähler manifold \(X\) is said to be of \(K3^{[m]}\)-type if it is deformation equivalent to the Hilbert scheme of \(m\) points on a \(K3\) surface. For \(K3^{[m]}\)-type, we have the following description of the Riemann–Roch polynomial (see [40]) \[\mathrm{RR}_{K3^{[m]}}(q) = \binom{\frac{1}{2} q + m+1}{m}.\] In particular, if \((X,H)\) is of \(K3^{[2]}\)-type with \(H\) very ample and of square \(2d\), assuming moreover that \(X\) is projectively normal when embedded in \(\mathbb{P}^n\), where \(n=\binom{ d+3}{2}-1\), we may compute the Hilbert function \[\label{eq:hilbert-function} \dim I_e = h^0(\mathbb{P}^n, \mathcal{I}_{X/\mathbb{P}^n}(e)) = h^0(\mathbb{P}^n,\mathcal{O}(e)) - h^0(X, H^e)= \binom{n+e}{e} - \binom{de^2+3}{2}.\tag{2}\] Notably, the number does not depend on the divisibility of \(H\).

Recall from the introduction that we denote by \(\mathcal{M}_{2d}^{(\gamma)}\) the moduli space of polarized hyperkähler manifolds \((X, H)\) of \(K3^{[2]}\)-type with square \(q(H)=2d\) and divisibility \(\mathrm{div}(H)=\gamma\). The space \(\mathcal{M}_{2d}^{(\gamma)}\) is non-empty if and only if \(\gamma=1\), or \(\gamma=2\) and \(2d\equiv 6\pmod 8\) [24]. In all cases, it is irreducible of dimension \(20\).

2.2 Syzygies and Green’s duality theorem↩︎

We recall some important notions in the study of syzygies for a projective variety. We will state the results in a limited generality that will be sufficient for the context of this paper. The definitive reference on the subject is of course Green’s influential paper [41], where one can find the original statements in their full generality.

Let \(X\) be a projective variety and let \(\varphi_H \colon X\mathchoice{\lhook\joinrel\longrightarrow}{\hookrightarrow}{\hookrightarrow}{\hookrightarrow}\mathbb{P}V^\vee\) be the embedding defined by a very ample line bundle \(H\) on \(X\), where \(V\mathrel{\vcenter{:}}= H^0(X, H)\). We denote by \(S_X\coloneq \mathop{\mathrm{Sym}}^\bullet V=\mathop{\mathrm{Sym}}^\bullet H^0(X,H)\) the symmetric algebra and by \(R_X\mathrel{\vcenter{:}}= \bigoplus_{d\ge 0} H^0(X,H^d)=\bigoplus_{d\ge 0} R_d\) the ring of sections.³ Clearly \(R_X\) admits the structure of a graded \(S_X\)-module via the natural map \(m:S_X\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow}R_X\). The embedding \(\varphi_H\) is said to be \(d\)-normal if \(m\) is surjective in degree \(d\); it is said to be projectively normal if it is \(d\)-normal for every \(d\).

By Hilbert’s syzygy theorem, there is a unique (up to isomorphism) minimal graded free resolution \[\dots\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow}F_2\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow}F_1\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow}F_0\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow}R_X\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow}0,\] where \[F_i=\bigoplus_j S_X(-i-j)^{b_{i,j}}\] are free \(S_X\)-modules with shifted degrees. The numbers \(b_{i,j}\) are called the Betti numbers and are invariants for the embedding \(\varphi_H\). It is customary to arrange them into the following Betti diagram, which provides a convenient way to visualize the properties of the embedding. \[\begin{array}{r|ccccc} & 0 & 1 & 2 & 3 & \dots \\ \hline 0 & b_{0,0} & b_{1,0} & b_{2,0} & b_{3,0} & \dots \\ 1 & b_{0,1} & b_{1,1} & b_{2,1} & b_{3,1} & \dots \\ 2 & b_{0,2} & b_{1,2} & b_{2,2} & b_{3,2} & \dots \\ \vdots & \vdots & \vdots & \vdots & \vdots & \ddots \\ \end{array}\] The Betti numbers can be effectively computed by means of the Koszul cohomology groups. Namely, one may consider the Koszul complex \[\cdots \mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow}{\mathchoice {\textstyle{\bigwedge\mkern-4.5mu^{i+1}\mkern 1mu}} {\textstyle{\bigwedge\mkern-4.5mu^{i+1}\mkern 1mu}} {\scriptstyle{\bigwedge\mkern-5mu^{i+1}}} {\scriptscriptstyle{\bigwedge\mkern-5mu^{i+1}}}} V\otimes R_{j-1}\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow}{\mathchoice {\textstyle{\bigwedge\mkern-4.5mu^{i}\mkern 1mu}} {\textstyle{\bigwedge\mkern-4.5mu^{i}\mkern 1mu}} {\scriptstyle{\bigwedge\mkern-5mu^{i}}} {\scriptscriptstyle{\bigwedge\mkern-5mu^{i}}}} V\otimes R_{j} \mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow}{\mathchoice {\textstyle{\bigwedge\mkern-4.5mu^{i-1}\mkern 1mu}} {\textstyle{\bigwedge\mkern-4.5mu^{i-1}\mkern 1mu}} {\scriptstyle{\bigwedge\mkern-5mu^{i-1}}} {\scriptscriptstyle{\bigwedge\mkern-5mu^{i-1}}}} V\otimes R_{j+1}\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow}\cdots\] and let \(\mathcal{K}_{i,j}\mathrel{\vcenter{:}}= \mathcal{K}_{i,j}(X,H)\) be the cohomogology group of the term in the middle. It holds that \(\dim \mathcal{K}_{i,j}=b_{i,j}\) [41].

Concerning varieties with trivial canonical bundle, we have the following result (see [loc. cit., Corollary 2.c.10]) that predicts some symmetry of the Betti diagram.

2.3 Moduli spaces of sheaves on \(K3\) surfaces↩︎

We recall some basic aspects of moduli spaces of stable sheaves on \(K3\) surfaces and their symplectic structure, a subject studied extensively by several authors, including Mukai, O’Grady, Huybrechts, and Yoshioka. The reader may consult [42] (and the references therein) for further details. For simplicity we will assume, as throughout the paper, that \((S,L)\) is a polarized \(K3\) surface with \(\mathop{\mathrm{Pic}}(S)=\mathbb{Z}\cdot L\).

Let \(\widetilde{H}(S,\mathbb{Z})\) denote the (even) cohomology ring of \(S\). The pure weight-two Hodge structure on \(H^2(S,\mathbb{Z})\) can be extended to \(\widetilde{H}(S,\mathbb{Z})\) by requiring the degree 0 and 4 parts to be algebraic. The symmetric bilinear form \[\langle u,v \rangle \mathrel{\vcenter{:}}= \int_S u_2\wedge v_2 - \int_S (u_0\wedge v_4 + u_4\wedge v_0)\] equips \(\widetilde{H}(S,\mathbb{Z})\) with the structure of a lattice, called the Mukai lattice of \(S\).

Given a coherent sheaf \(F\) on \(S\), we define its Mukai vector as \[v(F)\mathrel{\vcenter{:}}=\mathrm{ch}(F)\cdot\sqrt{\mathrm{td}(S)}=\left(\mathop{\mathrm{rk}}(F), c_{1}(F), \mathop{\mathrm{ch}}_2(F)+\mathop{\mathrm{rk}}(F)\right) \in \widetilde{H}(S,\mathbb{Z}).\] The Hirzebruch–Riemann–Roch formula implies that for coherent sheaves \(F,G\) on \(S\) we have \[\chi(E,F)=-\langle v(E),v(F)\rangle.\]

Let \(v=(r,cL,s) \in \widetilde{H}^{1,1}(S, \mathbb{Z})\) be a primitive vector with \(r>0\). We denote by \(\mathcal{M}(v)\) the moduli space of \(L\)-Gieseker stable sheaves on \(S\) with Mukai vector \(v\). It is known that \(\mathcal{M}(v)\) is non-empty exactly when \(v^2\geq -2\); in that case, \(\mathcal{M}(v)\) is a projective hyperkähler manifold of \(K3^{[m]}\)-type, where \(m=\frac{1}{2}(v^2+2)\).

The Beauville–Bogomolov–Fujiki form for \(\mathcal{M}(v)\) can be explicitly described as follows. There exists a tautological quasi-family of sheaves in \(\mathcal{M}(v)\) of similitude \(\rho\in\mathbb{Z}_{>0}\), that is, a sheaf \(\mathcal{E}\) on \(S\times \mathcal{M}(v)\) flat over \(\mathcal{M}(v)\) such that \(\mathcal{E}|_{S\times [F]}\cong F^{\oplus \rho}\) for every \([F]\in \mathcal{M}(v)\). For \(v^2\geq2\), letting \(\theta_{v}\colon \widetilde{H}(S,\mathbb{Z})\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow}H^2(\mathcal{M}(v),\mathbb{Z})\) be defined by \[\theta_{v}((\alpha_0,\alpha_2,\alpha_4))\mathrel{\vcenter{:}}= \frac{1}{\rho}\left[ \mathrm{pr}_{\mathcal{M}(v),!} \left(\mathrm{ch}(\mathcal{E})\smile\mathrm{pr}_S^{*}(\sqrt{\mathrm{td}(S)}\smile(\alpha_0,-\alpha_2,\alpha_4))\right)\right]_2,\] it was proved in [43], [44] that restriction to \(v^\perp\) yields a Hodge isometry \[\label{thetaiso} \theta_{v}\colon v^{\perp}\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow}H^2(\mathcal{M}(v),\mathbb{Z}).\tag{3}\]

For \(m\geq2\), the moduli space \(\mathcal{M}(1,0,-m+1)\) parametrizes ideal sheaves \(\mathcal{I}_Z\) where \(Z\) is a length-\(m\) 0-dimensional subscheme of \(S\), namely \(S^{[m]}\mathrel{\vcenter{:}}= \mathcal{M}(1,0,-m+1)\) is the Hilbert scheme of \(m\) points on \(S\). In that case, as proved in [1], 3 reads as a Hodge isometry \[\left(H^2(S^{[m]},\mathbb{Z}),q\right)\cong H^2(S,\mathbb{Z})\overset{\perp}{\oplus}\mathbb{Z}\cdot\delta\] where \(\delta\) is a \((1,1)\)-class of square \(-2(m-1)\). In particular \(\mathop{\mathrm{Pic}}(S^{[m]})\cong\mathop{\mathrm{Pic}}(S)\oplus\mathbb{Z}\cdot\delta\), where \(2\delta\) is the class of the divisor \(\Delta\) of non-reduced subschemes. For \(L\in \mathop{\mathrm{Pic}}(S)\), the corresponding line bundle on \(S^{[m]}\) is obtained by symmetrization and will be denoted by \(L_m\in\mathop{\mathrm{Pic}}(S^{[m]})\).

In the sequel, we will focus on the case \(m=2\). Consider the diagram \[\label{eq:universallenght2subscheme} \begin{tikzcd} &B\ar[ld, "\pi"']\ar[rd, "\psi"]\mathrel{\vcenter{:}}={\left\{{(p,\xi)\in S\times S^{[2]}}:{p\in \xi}\right\}}\\ S^{[2]}&& S \end{tikzcd}\tag{4}\] where \(B\) is the universal length-\(2\) subscheme (namely, \(\pi\) is a double cover branched along \(\Delta\)). For any sheaf \(F\) on \(S\) we can associate a tautological sheaf \(F^{[2]}\) on \(S^{[2]}\) by the rule \[F^{[2]} \mathrel{\vcenter{:}}= \pi_*\psi^*F.\] If \(F\) is a vector bundle of rank \(r\), then the corresponding tautological sheaf is a vector bundle of rank \(2r\), whose fiber at a point \(\xi\in S^{[2]}\) corresponds to \(H^0(S,F\otimes\mathcal{O}_\xi)\).

Recall that under the assumption \(\mathop{\mathrm{Pic}}(S)=\mathbb{Z}\cdot L\) and \(g\geq5\), then \(S\subset \mathbb{P}(H^0(L)^\vee)=\mathbb{P}^g\) is cut out by quadrics [7]. There is a natural identification \(H^0(S^{[2]},L_2-2\delta)\cong H^0(\mathbb{P}^g,\mathcal{I}_{S/\mathbb{P}^g}(2))\) with the space of quadratic equations for \(S\) in \(\mathbb{P}^g\), and the resulting morphism \[\varphi\colon S^{[2]}\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow}|L_2-2\delta|^\vee\] associates to each \(\xi\in S^{[2]}\) the hyperplane of quadric hypersurfaces in \(\mathbb{P}^g\) containing both \(S\) and the line spanned by \(\xi\). Note that the morphism \(\varphi\) is indeed well-defined, as \(S\) is cut out by quadrics and contains no line. We have a canonical short exact sequence \[0\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow}K \mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow}H^0\left(\mathcal{I}_{S/\mathbb{P}}(2)\right)\otimes\mathcal{O}_{\mathbb{P}}\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow}\mathcal{N}^\vee_{S/\mathbb{P}}(2)\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow}0,\] where \(K\) is an extension of \(\mathcal{I}_{X/\mathbb{P}}^2(2)\) by \(\ker\bigl(H^0\left(\mathcal{I}_{S/\mathbb{P}}(2)\right)\otimes\mathcal{O}_{\mathbb{P}}\twoheadrightarrow \mathcal{I}_{S/\mathbb{P}}(2)\bigr)\). It follows that \(H^0(K)=0\), hence there is a canonical inclusion \[\label{quadrics-conormal} H^0\left(\mathcal{I}_{S/\mathbb{P}}(2)\right)\mathchoice{\lhook\joinrel\longrightarrow}{\hookrightarrow}{\hookrightarrow}{\hookrightarrow}H^0\left(\mathcal{N}^\vee_{S/\mathbb{P}}(2)\right)\tag{5}\] such that, for a given quadric \(Q\in H^0\left(\mathcal{I}_{S/\mathbb{P}}(2)\right)\), the zero locus of the induced global section of the twisted conormal bundle \(\mathcal{N}^\vee_{S/\mathbb{P}}(2)\) equals \(S\cap\mathop{\mathrm{Sing}}(Q)\).

2.4 Quadrics of low rank containing \(K3\) surfaces↩︎

In this subsection, we recall a few aspects of the locus of quadrics of rank \(\leq 6\) in \(|I_S(2)|\mathrel{\vcenter{:}}= \mathbb{P}H^0(\mathcal{I}_{S/\mathbb{P}^g}(2))\). As it is well known to the experts, this locus can be described in terms of stable rank \(2\) sheaves on the \(K3\) surface \(S\). We provide some details on the description of this locus following the lines of [28], where O’Grady focuses on a particular component \(Y_0\).

Consider a pair \((E,V)\), for a rank-\(2\) stable sheaf \(E\in\mathcal{M}(2,L,\ell+2)\) (with \(0\leq \ell \leq \lfloor\frac{g}{2}\rfloor-2\)) and a 4-dimensional subspace \(V\subset H^0(E)\). Associated to it we have a rational map \[\rho_{E,V}\colon S\dasharrow \mathop{\mathrm{Gr}}(2,V^\vee)\subset \mathbb{P}({\mathchoice {\textstyle{\bigwedge\mkern-4.5mu^{2}\mkern 1mu}} {\textstyle{\bigwedge\mkern-4.5mu^{2}\mkern 1mu}} {\scriptstyle{\bigwedge\mkern-5mu^{2}}} {\scriptscriptstyle{\bigwedge\mkern-5mu^{2}}}} V^\vee),\quad p\mathchoice{\longmapsto}{\mapsto}{\mapsto}{\mapsto}\left(V\cap H^0(E\otimes\mathcal{I}_p)\right)^\perp.\] Note that \(\rho_{E,V}\) is precisely not defined at the finite set of points \(p\in S\) where either \(E\) is not locally free, or \(E\) is locally free but \(V\cap H^0(E\otimes\mathcal{I}_p)\) has codimension \(<2\) in \(V\). Moreover, over the open subset \(U\subset S\) where \(\rho_{E,V}\) is defined, it is straightforward to check that \(\rho_{E,V}^*\mathcal{U}^\vee\cong E|_U\), where \(\mathcal{U}\) is the universal subbundle of rank \(2\).

The natural map \(\lambda_{E,V}\colon{\mathchoice {\textstyle{\bigwedge\mkern-4.5mu^{2}\mkern 1mu}} {\textstyle{\bigwedge\mkern-4.5mu^{2}\mkern 1mu}} {\scriptstyle{\bigwedge\mkern-5mu^{2}}} {\scriptscriptstyle{\bigwedge\mkern-5mu^{2}}}} V\mathchoice{\lhook\joinrel\longrightarrow}{\hookrightarrow}{\hookrightarrow}{\hookrightarrow}{\mathchoice {\textstyle{\bigwedge\mkern-4.5mu^{2}\mkern 1mu}} {\textstyle{\bigwedge\mkern-4.5mu^{2}\mkern 1mu}} {\scriptstyle{\bigwedge\mkern-5mu^{2}}} {\scriptscriptstyle{\bigwedge\mkern-5mu^{2}}}} H^0(E)\overset\det\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow}H^0(L)\) yields a factorization \[\label{eq:diagram} \begin{tikzcd} S\ar[rd,dashed,"\rho_{E,V}"']\ar[r,hookrightarrow]&\mathbb{P}\left(H^0(L)^\vee\right)\ar[r,dashed]&\mathbb{P}\left(\mathrm{Im}(\lambda_{E,V})^\vee\right) \ar[d,hookrightarrow,"\mathbb{P}(\lambda_{E,V}^\vee)"] \\ &\mathop{\mathrm{Gr}}(2,V^\vee)\ar[r,hookrightarrow]&\mathbb{P}\left({\mathchoice {\textstyle{\bigwedge\mkern-4.5mu^{2}\mkern 1mu}} {\textstyle{\bigwedge\mkern-4.5mu^{2}\mkern 1mu}} {\scriptstyle{\bigwedge\mkern-5mu^{2}}} {\scriptscriptstyle{\bigwedge\mkern-5mu^{2}}}} V^\vee\right) \end{tikzcd}\tag{6}\]

where the rational map in the first row is the linear projection from the annihilator subspace \(\mathbb{P}\left(\mathrm{Im}(\lambda_{E,V})^\perp\right)\subset \mathbb{P}\left(H^0(L)^\vee\right)\). In other words, \(\mathbb{P}(\mathrm{Im}(\lambda_{E,V})^\vee)\) is the linear span of the image \(\rho_{E,V}(S)\).

The Grassmannian \(\mathop{\mathrm{Gr}}(2,V^\vee)\subset \mathbb{P}({\mathchoice {\textstyle{\bigwedge\mkern-4.5mu^{2}\mkern 1mu}} {\textstyle{\bigwedge\mkern-4.5mu^{2}\mkern 1mu}} {\scriptstyle{\bigwedge\mkern-5mu^{2}}} {\scriptscriptstyle{\bigwedge\mkern-5mu^{2}}}}V^\vee)\) is a smooth quadric. Therefore, as long as it does not contain the linear span of \(\rho_{E,V}(S)\), the pullback defines a quadric \(Q_{E,V}\subset \mathbb{P}\left(H^0(L)^\vee\right)\) containing \(S\). And this is indeed the case by the following lemma.

By construction, the singular locus of \(Q_{E,V}\) is \(\mathbb{P}\left(\mathrm{Im}(\lambda_{E,V})^\perp\right)\subset \mathbb{P}\left(H^0(L)^\vee\right)\), the linear subspace we are projecting from. We see that the quadric has either rank \(5\) or \(6\).

Performing this construction in families, we obtain for every \(\ell\) (\(0\leq \ell \leq \lfloor\frac{g}{2}\rfloor-2\)) a natural morphism \[\label{eq:def-relGrass} \psi_{\ell}\colon \mathcal{G}_\ell\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow}{\left\{{Q\in|I_S(2)|}:{\mathop{\mathrm{rk}}(Q)\leq 6}\right\}}\subset |I_S(2)|,\quad(E,V)\longmapsto Q_{E,V}\tag{7}\] where \(\mathcal{G}_\ell\) is the relative Grassmannian with fiber \(\mathop{\mathrm{Gr}}(4,H^0(E))\) over \(E\in\mathcal{M}(2,L,\ell+2)\). Note that \(\mathcal{G}_\ell\) are all irreducible of dimension \(2g-8\). We will denote the image by \(Y_\ell\mathrel{\vcenter{:}}=\mathop{\mathrm{Im}}(\psi_\ell)\subset |I_S(2)|\).

The following lemma shows that, via the morphisms \(\psi_\ell\), vector bundles of rank \(2\) serve as a bridge between quadrics of rank \(\leq 6\) in \(|I_S(2)|\) and secant lines to \(S\). This will be essential in our analysis of the linear system \(|L_2-2\delta|\) in \(S^{[2]}\):

It is a natural question whether the varieties \(Y_\ell\) recover all quadrics of rank \(\leq 6\) containing \(S\). To this end, we will provide an intrinsic description for \(Y_\ell\) in terms of the quadrics they contain. Consider the following locally closed subset of \(|I_S(2)|\): \[\Sigma\mathrel{\vcenter{:}}={\left\{{Q\in|I_S(2)|}:{\mathop{\mathrm{rk}}(Q)=6,\; \mathop{\mathrm{Sing}}(Q)\cap S=\varnothing}\right\}}.\]

For \(Q\in \Sigma\), the projection from the singular locus of \(Q\) defines a morphism \(\rho\colon S\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow}\mathbb{P}^5\) whose image is contained in a smooth quadric \(Q_4\subset\mathbb{P}^5\). It is well known that \(H^4(Q_4,\mathbb{Z})\) is spanned by the classes \(P_1\) and \(P_2\) of the two types of 2-planes contained in \(Q_4\), and \(h^2=P_1+P_2\) where \(h=c_1(\mathcal{O}_{Q_4}(1))\). Hence, there exist \(a,b\in \mathbb{Z}_{\ge0}\) with \[[\rho(S)]=a P_1 + b P_2,\,\;a+b=\deg S=2g-2.\] Therefore, we have the following invariant of a quadric \(Q\in\Sigma\) which is locally constant on \(\Sigma\) \[i(Q)\mathrel{\vcenter{:}}=\tfrac{1}{2}\bigl\lvert a-b\bigr\rvert\in\left\{0,\dots,g-1\right\}.\] We can thus define \[\Sigma_\ell\mathrel{\vcenter{:}}={\left\{{Q\in \Sigma}:{i(Q)=\ell}\right\}},\] and immediately obtain a decomposition \[\Sigma=\Sigma_0\sqcup \dots\sqcup \Sigma_{g-1}.\]

3 Proof of 1↩︎

3.1 Very ampleness for \(g\geq 7\)↩︎

Let \((S,L)\) be a polarized \(K3\) surface of genus \(g\), with \(\mathop{\mathrm{Pic}}(S)=\mathbb{Z}\cdot L\). Given a stable vector bundle \(E\) with \(v(E)=(2,L,\lfloor\frac{g}{2}\rfloor)\), we consider the rational map \[\varphi_E\colon S^{[2]}\dasharrow \mathop{\mathrm{Gr}}\left(\lfloor\tfrac{g}{2}\rfloor-2, H^0(E)\right)\mathchoice{\lhook\joinrel\longrightarrow}{\hookrightarrow}{\hookrightarrow}{\hookrightarrow}\mathbb{P}\left({\mathchoice {\textstyle{\bigwedge\mkern-4.5mu^{\lfloor\frac{g}{2}\rfloor-2}\mkern 1mu}} {\textstyle{\bigwedge\mkern-4.5mu^{\lfloor\frac{g}{2}\rfloor-2}\mkern 1mu}} {\scriptstyle{\bigwedge\mkern-5mu^{\lfloor\frac{g}{2}\rfloor-2}}} {\scriptscriptstyle{\bigwedge\mkern-5mu^{\lfloor\frac{g}{2}\rfloor-2}}}}H^0(E)\right)\] sending a general \(\xi\in S^{[2]}\) to the codimension \(4\) subspace \(H^0(E\otimes\mathcal{I}_\xi)\subset H^0(E)\). The map \(S^{[2]}\dasharrow \mathop{\mathrm{Gr}}\left(\lfloor\frac{g}{2}\rfloor-2, H^0(E)\right)\) is naturally obtained from the evaluation map \[H^0(E)\otimes\mathcal{O}_{S^{[2]}}\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow}E^{[2]},\] where \(E^{[2]}\) is the tautological rank \(4\) bundle on \(S^{[2]}\) induced by \(E\). In particular, \[\varphi_E^*\mathcal{O}(1)=\det(E^{[2]})=L_2-2\delta.\]

Note that the vector bundle \(E\) is unique when \(g\) is even. In that case, the following result is enough to conclude the very ampleness of the complete linear system \(|L_2-2\delta|\):

If \(g\) is odd, then the moduli space \(\mathcal{M}\mathrel{\vcenter{:}}=\mathcal{M}(2,L,\lfloor\frac{g}{2}\rfloor)\) is 2-dimensional. In that case we can combine all the rational maps \(\varphi_E\) (with \(E\) varying in \(\mathcal{M}\)) to prove:

3.2 Strange duality↩︎

In this subsection we show Le Potier’s strange duality conjecture for the pair \((S^{[2]},L_2-2\delta)\) for any \(g\geq6\). This generalizes a result by O’Grady [28], who established the conjecture in the range \(6\leq g\leq 8\).

Strange duality is a statement about duality of sections of determinant line bundles on moduli spaces of sheaves. Whereas we refer the reader to [49] for a gentle introduction to the subject and further details, here we restrict ourselves to the Mukai vectors \(v\mathrel{\vcenter{:}}=(1,0,-1)\) and \(w\mathrel{\vcenter{:}}=(2,L,2)\) on \(S\).

Note that \(v^2=2\) and \(w^2=2g-10\geq2\). According to 3 , we have isometries \[\theta_v\colon v^\perp\overset{\cong}{\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow}} \mathop{\mathrm{Pic}}(\mathcal{M}(v))=\mathop{\mathrm{Pic}}(S^{[2]}),\quad \theta_w\colon w^\perp\overset{\cong}{\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow}} \mathop{\mathrm{Pic}}(\mathcal{M}(w))\] defined via determinant line bundles. In particular, since \(\langle v,w\rangle =0\), we have an induced line bundle \(H\mathrel{\vcenter{:}}=\theta_w(v)\) on \(\mathcal{M}(w)\). Furthermore, in virtue of 1 the equality \(L_2-2\delta=\theta_v(w)\) holds.

We have \(h^2(E\otimes\mathcal{I}_\xi)=0\) for every \(E\in\mathcal{M}(w)\) and \(\xi\in S^{[2]}\), and the jump locus \[\Theta\mathrel{\vcenter{:}}=\bigl\{(E,\xi)\in\mathcal{M}(w)\times S^{[2]}: h^1(E\otimes\mathcal{I}_\xi)>0\bigr\}\] defines a divisor in the product \(\mathcal{M}(w)\times S^{[2]}\), corresponding to the line bundle \(H\boxtimes (L_2-2\delta)\). Therefore, we have a natural (up to constant) morphism \[\label{SD} H^0(S^{[2]},L_2-2\delta)^\vee\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow}H^0(\mathcal{M}(w),H).\tag{10}\] The strange duality conjecture predicts the following:

In the range \(6\leq g \leq 8\), O’Grady [28] proved strange duality employing the following arguments:

1. We have \(h^0(S^{[2]}, L_2-2\delta)=\binom{g-2}{2}\) (as \(S\subset \mathbb{P}^g\) is projectively normal) and \(h^0(\mathcal{M}(w),H)=\chi(\mathcal{M}(w),H)=\binom{g-2}{2}\) (as \(H\) is ample for \(g\leq 8\)).

2. Let \(f\colon\mathcal{M}(2,L,2)\dasharrow |I_S(2)|=\mathbb{P}(H^0(L_2-2\delta))\) be the rational map induced by \(\psi_0\) in 7 . Then \(f\) is a morphism, and \(f^*\mathcal{O}(1)=H\).

3. \(Y_0=\mathop{\mathrm{Im}}(f)\subset |I_S(2)|\) is a non-degenerate subvariety. Therefore, \(f^*\) induces an injection \[|I_S(2)|^\vee\mathchoice{\lhook\joinrel\longrightarrow}{\hookrightarrow}{\hookrightarrow}{\hookrightarrow}|H|\] which must be an isomorphism, by the equality of dimensions in [SD-cond1].

4. The pullback map \(f^*\colon |I_S(2)|^\vee\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow}|H|\) is the projectivization of 10 .

In the case of an arbitrary \(g\geq 6\), we observe that:

1. The equalities \(h^0(S^{[2]}, L_2-2\delta)=\binom{g-2}{2}=h^0(\mathcal{M}(w),H)\) still hold, as \(H\) is big and nef for \(g\geq 9\) (see for instance [24]).

2. For \(g\geq9\) the map \(f\) is not regular, with indeterminacy given by the Brill–Noether locus \[\{E\in\mathcal{M}(w):\;h^1(E)>0\}\] which has codimension 5 in \(\mathcal{M}(w)\) (see [50] for a more detailed description). The argument in [28] still shows \(f^*\mathcal{O}(1)=H\) in this case.

3. The induced map \(f^*\colon |I_S(2)|^\vee\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow}|H|\) is still the projectivization of 10 . This can be argued exactly as in [28], by using 2 (which generalizes Claim 5.17 in loc. cit.).

Therefore, in order to establish 1.[thm:L-2delta-very-ample-2] it suffices to prove [SD-cond3] for any \(g\geq6\), namely that \(Y_0\subset |I_S(2)|\) is non-degenerate. Note that the argument in [28] cannot be applied, as it relies on an explicit description of \(|I_S(2)|\) in terms of the Mukai model of \(S\).

To circumvent this, we consider a general curve \(C\in|L|\), which is Brill–Noether–Petri general by [51]. Consider the cartesian diagram \[\begin{tikzcd} C \arrow[hookrightarrow,"i"]{d}\arrow[hookrightarrow]{r} & \mathbb{P}^{g-1}=\mathbb{P}(H^0(\omega_C)^\vee) \arrow[hookrightarrow]{d} \\ S \arrow[hookrightarrow]{r} & \mathbb{P}^{g}=\mathbb{P}(H^0(L)^\vee) \; \end{tikzcd}\] inducing a natural identification of quadrics \(|I_{C/\mathbb{P}^{g-1}}(2)|=|I_{S}(2)|\). According to a theorem of Green [29], \(|I_{C/\mathbb{P}^{g-1}}(2)|\) is spanned by a (\(g-4\))-dimensional subvariety \(\mathcal{Z}\) parametrizing certain quadrics of rank \(\leq 4\). \(\mathcal{Z}\) is birational to the quotient \(W^1_{g-1}(C)/\langle \iota\rangle\), where \(\iota\) is the involution given by \(\iota(A)=\omega_C\otimes A^{-1}\).

The general element of \(\mathcal{Z}\) is given as follows. Fix \(A\in W^1_{g-1}(C)\) with \(h^0(A)=2\), such that both \(A\) and \(\omega_C\otimes A^{-1}\) are basepoint-free. Then the Segre quadric (parametrizing decomposable tensors in \(\mathbb{P}\left(H^0(A)^\vee\otimes H^0(\omega_C\otimes A^{-1})^\vee\right)\)) induces via the Petri map \[m_A:H^0(A)\otimes H^0(\omega_C\otimes A^{-1})\hookrightarrow H^0(\omega_C)\] a quadric \(Q_A\) of rank \(\leq 4\) in \(\mathbb{P}(H^0(\omega_C)^\vee)\) containing the canonical curve \(C\). In coordinates: if \(s_1,s_2\) and \(t_1,t_2\) denote respective bases of \(H^0(A)\) and \(H^0(\omega_C\otimes A^{-1})\), then \[Q_A=\det\begin{pmatrix} m_A(s_1\otimes t_1) & m_A(s_1\otimes t_2) \\ m_A(s_2\otimes t_1) & m_A(s_2\otimes t_2) \end{pmatrix}\in \mathop{\mathrm{Sym}}^2 H^0(C,\omega_C).\]

In view of Green’s theorem, non-degeneracy of \(Y_0\) in \(|I_S(2)|\) is implied by the following:

4 Genus 7↩︎

4.1 Quadrics containing the Mukai model↩︎

Let \((S,L)\) be a polarized \(K3\) surface of genus \(7\) with \(\mathop{\mathrm{Pic}}(S)=\mathbb{Z}\cdot L\). We first give a brief description of the Mukai model for the embedding \(S\mathchoice{\lhook\joinrel\longrightarrow}{\hookrightarrow}{\hookrightarrow}{\hookrightarrow}\mathbb{P}(H^0(L)^\vee)=\mathbb{P}^7\), following [2].

Let \(E\) be the unique stable vector bundle with Mukai vector \(v(E)=(5,2L,5)\). The evaluation map of global sections fits in a short exact sequence \[\label{globgeneration7} 0\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow}E^\vee \mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow}H^0(E)\otimes\mathcal{O}_S\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow}E\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow}0,\tag{13}\] and defines an immersion \(S\mathchoice{\lhook\joinrel\longrightarrow}{\hookrightarrow}{\hookrightarrow}{\hookrightarrow}\mathop{\mathrm{Gr}}(5, H^0(E)^\vee)\) by the rule \(p\mathchoice{\longmapsto}{\mapsto}{\mapsto}{\mapsto}H^0(E\otimes\mathcal{I}_p)^\perp\).

By [2], the 10-dimensional vector space \(H^0(E)^\vee\) is equipped with a canonical (up to scalar) non-degenerate symmetric bilinear form \(q\), corresponding to the kernel of the natural map \[\mathop{\mathrm{Sym}}^2H^0(E)\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow}H^0(\mathop{\mathrm{Sym}}^2E).\] By construction, \(H^0(E\otimes\mathcal{I}_p)^\perp\subset H^0(E)^\vee\) is an isotropic subspace with respect to \(q\) for all \(p\in S\). In this way, \(S\) is mapped into \(\mathop{\mathrm{OGr}}(5,H^0(E)^\vee)_+\), one of the two connected components of the \(10\)-dimensional orthogonal Grassmannian parametrizing isotropic subspaces with respect to \(q\).

The Plücker line bundle on \(\mathop{\mathrm{OGr}}(5,H^0(E)^\vee)_+\) coming from \(\mathop{\mathrm{OGr}}(5,H^0(E)^\vee)_+\mathchoice{\lhook\joinrel\longrightarrow}{\hookrightarrow}{\hookrightarrow}{\hookrightarrow}\mathbb{P}({\mathchoice {\textstyle{\bigwedge\mkern-4.5mu^{5}\mkern 1mu}} {\textstyle{\bigwedge\mkern-4.5mu^{5}\mkern 1mu}} {\scriptstyle{\bigwedge\mkern-5mu^{5}}} {\scriptscriptstyle{\bigwedge\mkern-5mu^{5}}}}H^0(E)^\vee)\) is not primitive: it equals twice a very ample line bundle \(\mathcal{O}_{\mathop{\mathrm{OGr}}_+}(1)\) inducing the spinor embedding \[\mathop{\mathrm{OGr}}(5,H^0(E)^\vee)_+\mathchoice{\lhook\joinrel\longrightarrow}{\hookrightarrow}{\hookrightarrow}{\hookrightarrow}\mathbb{P}^{15}=\mathbb{P}V_+.\] The \(K3\) surface \(S\mathchoice{\lhook\joinrel\longrightarrow}{\hookrightarrow}{\hookrightarrow}{\hookrightarrow}\mathbb{P}(H^0(L)^\vee)=\mathbb{P}^7\) is then a \(\mathbb{P}^7\)-linear section of \(\mathop{\mathrm{OGr}}(5,H^0(E)^\vee)_+\).

For later use, let us give some details on the spinor embedding of \(\mathop{\mathrm{OGr}}(5,H^0(E)^\vee)_+\). We follow the notations of [53]. Say \(V=H^0(E)^\vee\), and pick two disjoint isotropic subspaces \(U_0,U_\infty\subset V\), so that \(V\cong U_0\oplus U_\infty\) and \(U_0\cong U_\infty^\vee\). There is a linear map \[V\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow}\mathrm{End}({\mathchoice {\textstyle{\bigwedge\mkern-4.5mu^{\bullet }\mkern 1mu}} {\textstyle{\bigwedge\mkern-4.5mu^{\bullet }\mkern 1mu}} {\scriptstyle{\bigwedge\mkern-5mu^{\bullet }}} {\scriptscriptstyle{\bigwedge\mkern-5mu^{\bullet }}}}U_\infty),\quad v\mathchoice{\longmapsto}{\mapsto}{\mapsto}{\mapsto}\varphi_v\] where \(\varphi_v\) is defined by wedge (resp. by derivation) if \(v\in U_\infty\) (resp. if \(v\in U_0=U_\infty^\vee\)). If we are taking \(U_0\in\mathop{\mathrm{OGr}}(5,H^0(E)^\vee)_+\), then the spinor embedding \[\mathop{\mathrm{OGr}}(5,H^0(E)^\vee)_+\mathchoice{\lhook\joinrel\longrightarrow}{\hookrightarrow}{\hookrightarrow}{\hookrightarrow}\mathbb{P}^{15}=\mathbb{P}\left({\mathchoice {\textstyle{\bigwedge\mkern-4.5mu^{\mathrm{even}}\mkern 1mu}} {\textstyle{\bigwedge\mkern-4.5mu^{\mathrm{even}}\mkern 1mu}} {\scriptstyle{\bigwedge\mkern-5mu^{\mathrm{even}}}} {\scriptscriptstyle{\bigwedge\mkern-5mu^{\mathrm{even}}}}} U_\infty\right),\quad U\mathchoice{\longmapsto}{\mapsto}{\mapsto}{\mapsto}[s_U]\] sends an isotropic subspace \(U\) to the unique (up to constant) spinor \(s_U\) with the property \(\varphi_u(s_U)=0\) for every \(u\in U\). For instance, \(s_{U_0}=1\in {\mathchoice {\textstyle{\bigwedge\mkern-4.5mu^{0}\mkern 1mu}} {\textstyle{\bigwedge\mkern-4.5mu^{0}\mkern 1mu}} {\scriptstyle{\bigwedge\mkern-5mu^{0}}} {\scriptscriptstyle{\bigwedge\mkern-5mu^{0}}}} U_\infty\). Similarly, the other component of the orthogonal Grassmannian admits an embedding \[\mathop{\mathrm{OGr}}(5,H^0(E)^\vee)_-\mathchoice{\lhook\joinrel\longrightarrow}{\hookrightarrow}{\hookrightarrow}{\hookrightarrow}(\mathbb{P}^{15})^\vee=\mathbb{P}\left({\mathchoice {\textstyle{\bigwedge\mkern-4.5mu^{\mathrm{odd}}\mkern 1mu}} {\textstyle{\bigwedge\mkern-4.5mu^{\mathrm{odd}}\mkern 1mu}} {\scriptstyle{\bigwedge\mkern-5mu^{\mathrm{odd}}}} {\scriptscriptstyle{\bigwedge\mkern-5mu^{\mathrm{odd}}}}} U_\infty\right).\]

Regarding \(S\) as a codimension \(8\) linear section of \(\mathop{\mathrm{OGr}}(5,H^0(E)^\vee)_+\), there is an identification of \(|I_{S}(2)|\) with \(\mathbb{P}H^0\left(\mathcal{I}_{\mathop{\mathrm{OGr}}_+/\mathbb{P}^{15}}(2)\right)\); the latter can be identified with \(\mathbb{P}(H^0(E)^\vee)\) as follows. Let \(\kappa\colon{\mathchoice {\textstyle{\bigwedge\mkern-4.5mu^{\bullet }\mkern 1mu}} {\textstyle{\bigwedge\mkern-4.5mu^{\bullet }\mkern 1mu}} {\scriptstyle{\bigwedge\mkern-5mu^{\bullet }}} {\scriptscriptstyle{\bigwedge\mkern-5mu^{\bullet }}}}U_\infty\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow}{\mathchoice {\textstyle{\bigwedge\mkern-4.5mu^{5}\mkern 1mu}} {\textstyle{\bigwedge\mkern-4.5mu^{5}\mkern 1mu}} {\scriptstyle{\bigwedge\mkern-5mu^{5}}} {\scriptscriptstyle{\bigwedge\mkern-5mu^{5}}}} U_\infty\cong \mathbb{C}\) denote the projection onto the top part, and consider the bilinear form \(\beta\) on \({\mathchoice {\textstyle{\bigwedge\mkern-4.5mu^{\bullet }\mkern 1mu}} {\textstyle{\bigwedge\mkern-4.5mu^{\bullet }\mkern 1mu}} {\scriptstyle{\bigwedge\mkern-5mu^{\bullet }}} {\scriptscriptstyle{\bigwedge\mkern-5mu^{\bullet }}}}U_\infty\) defined by the rule \[\beta(\xi,\xi')=(-1)^{\frac{p(p+1)}{2}}\kappa(\xi\wedge\xi'),\] for \(\xi\) homogeneous of degree \(p\). Then for every \(v\in V\), the rule \(N_v(s)=\beta(s,\varphi_v(s))\) defines a quadratic form \(N_v\) on \({\mathchoice {\textstyle{\bigwedge\mkern-4.5mu^{\mathrm{even}}\mkern 1mu}} {\textstyle{\bigwedge\mkern-4.5mu^{\mathrm{even}}\mkern 1mu}} {\scriptstyle{\bigwedge\mkern-5mu^{\mathrm{even}}}} {\scriptscriptstyle{\bigwedge\mkern-5mu^{\mathrm{even}}}}} U_\infty\), such that \(H^0\left(\mathcal{I}_{\mathop{\mathrm{OGr}}_+/\mathbb{P}^{15}}(2)\right)=\{N_v\}_{v\in V}\).

4.2 The Fourier–Mukai partner↩︎

The geometry of the Mukai model for \(S\) is very much related to the Mukai model for the (fine) moduli space \(\mathcal{M}\mathrel{\vcenter{:}}=\mathcal{M}(2,L,3)\), which is another \(K3\) surface whose Picard group is generated by a polarization \(\widehat{L}\) of genus 7. Let us give a brief account; the reader is referred to [54], [55] for details.

Let \(\mathcal{F}\) denote a universal rank \(2\) bundle on \(S\times\mathcal{M}\); for an appropriate choice (obtained by twisting \(\mathcal{F}\) with the pullback of a line bundle on \(\mathcal{M}\)) \(\mathcal{F}\) also becomes the universal family of vector bundles on \(\mathcal{M}\) with Mukai vector \((2,\widehat{L},3)\). Given a point \(p\in S\), we will denote by \(\mathcal{F}_p\) the corresponding vector bundle on \(\mathcal{M}\).

Denote by \(p_S\) and \(p_\mathcal{M}\) the respective projections of \(S\times \mathcal{M}\) onto \(S\) and \(\mathcal{M}\). Then the rank \(5\) bundle \(E\) on \(S\) can be recovered as \(p_{S*}\mathcal{F}\), in particular \(H^0(E)=H^0(\mathcal{F})\). By symmetry \(E_\mathcal{M}\mathrel{\vcenter{:}}= p_{\mathcal{M}*}\mathcal{F}\) is the stable vector bundle with Mukai vector \((5,2\widehat{L},5)\), and the immersion \(\mathcal{M}\mathchoice{\lhook\joinrel\longrightarrow}{\hookrightarrow}{\hookrightarrow}{\hookrightarrow}\mathop{\mathrm{Gr}}(5, H^0(E)^\vee)\) factors through the other component \(\mathop{\mathrm{OGr}}(5,H^0(E)^\vee)_-\) of the orthogonal Grassmannian. If we consider the spinor embedding of \(\mathop{\mathrm{OGr}}(5,H^0(E)^\vee)_-\) in the dual space \((\mathbb{P}^{15})^\vee\), then \(\mathcal{M}\) becomes the linear section of \(\mathop{\mathrm{OGr}}(5,H^0(E)^\vee)_-\) by the 7-dimensional linear space which is orthogonal to the linear span of \(S\) in \(\mathbb{P}^{15}\).

It is also worth mentioning that \((S^{[2]},L_2-2\delta)\cong(\mathcal{M}^{[2]},\widehat{L}_2-2\widehat\delta)\) as polarized varieties. Indeed the isomorphism \(S^{[2]}\cong\mathcal{M}^{[2]}\) can be realized by means of a Fourier–Mukai transform (see [48]), and a simple numerical computation shows that the polarization is preserved. Whereas we will provide another geometric interpretation for this isomorphism in 9, let us give some more details on its Fourier–Mukai realization. We denote by \[\Phi\mathrel{\vcenter{:}}=\Phi_\mathcal{F}^{S\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow}\mathcal{M}}\colon\mathop{\mathrm{D}^{\mathrm{b}}}\nolimits(S)\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow}\mathop{\mathrm{D}^{\mathrm{b}}}\nolimits(\mathcal{M}),\quad\widehat{\Phi}\mathrel{\vcenter{:}}=\Phi_\mathcal{F}^{\mathcal{M}\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow}S}\colon\mathop{\mathrm{D}^{\mathrm{b}}}\nolimits(\mathcal{M})\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow}\mathop{\mathrm{D}^{\mathrm{b}}}\nolimits(S)\] the Fourier–Mukai equivalences with kernel \(\mathcal{F}\). The isomorphism \(\iota\colon\mathcal{M}^{[2]}\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow}S^{[2]}\) is constructed in [48] by proving that, for a given \(\tau\in\mathcal{M}^{[2]}\), the derived dual \(\widehat{\Phi}(\mathcal{I}_\tau)^\vee\) is the ideal sheaf of a length-2 subscheme \(\xi\) on \(S\); one sets \(\iota(\tau)=\xi\). When \(\tau=\{F_1,F_2\}\) is reduced, the short exact sequence \[0\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow}\mathcal{I}_\tau\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow}\mathcal{O}_\mathcal{M}\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow}\mathcal{O}_\tau\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow}0\] on \(\mathcal{M}\) induces a short exact sequence \[0\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow}\widehat{\Phi}(\mathcal{O}_\tau)^\vee=F_1^\vee\oplus F_2^\vee\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow}\widehat{\Phi}(\mathcal{O}_\mathcal{M})^\vee=E^\vee\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow}\widehat{\Phi}(\mathcal{I}_\tau)^\vee=\mathcal{I}_{\iota(\tau)}\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow}0\] on \(S\) (similarly, if \(\tau\) is non-reduced and supported at \(F\in\mathcal{M}\), then \(\widehat{\Phi}(\mathcal{O}_\tau)^\vee\) is a self-extension of \(F^\vee\)). In this way, for a given \(\xi\in S^{[2]}\) the support of \(\iota^{-1}(\xi)\) consists of those vector bundles \(F\in\mathcal{M}\) with \(h^1(S,F\otimes\mathcal{I}_\xi)=1\) (or equivalently, \(h^0(S,F\otimes\mathcal{I}_\xi)=2\)); see also [47] for further details.

4.3 Geometry of the linear system \(|L_2-2\delta|\)↩︎

Given two distinct points \(p,q\in S\), following [53] the intersection \[H^0(E\otimes\mathcal{I}_p)^\perp\cap H^0(E\otimes\mathcal{I}_q)^\perp=\left(H^0(E\otimes\mathcal{I}_p)+H^0(E\otimes\mathcal{I}_q)\right)^\perp\] is odd-dimensional, hence so is the subspace \(H^0(E\otimes\mathcal{I}_{p,q})=H^0(E\otimes\mathcal{I}_p)\cap H^0(E\otimes\mathcal{I}_q)\). In particular, \(h^0(E\otimes\mathcal{I}_{p,q})\neq0\). More concretely, we have:

An immediate consequence of 3.[item1:sectionsE] is the existence of a natural morphism \[\phi\colon S^{[2]}\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow}\mathbb{P}(H^0(E)),\quad \xi\mathchoice{\longmapsto}{\mapsto}{\mapsto}{\mapsto}H^0(E\otimes\mathcal{I}_\xi).\]

In particular, \(\phi\) is a closed immersion by 1.[thm:L-2delta-very-ample-1]. Recalling that there is an isomorphism \(H^0(E)\cong H^0(E)^\vee\) provided by the bilinear form \(q\), and hence \(|I_S(2)|\cong |I_S(2)|^\vee\) canonically, we have a more geometric realization of the embedding \(\phi\):

By 3, we have a stratification \[S^{[2]}\mathchoice{\lhook\joinrel\longrightarrow}{\hookrightarrow}{\hookrightarrow}{\hookrightarrow}\Gamma\mathchoice{\lhook\joinrel\longrightarrow}{\hookrightarrow}{\hookrightarrow}{\hookrightarrow}\mathbb{P}(H^0(E))\] of the space of global sections according to the length of their zero locus (i.e. \(\Gamma\) parametrizes global sections with non-empty zero locus). This stratification fits in a commutative diagram \[\label{eq:diagram-genus7} \begin{tikzcd} B \ar[d,"\pi"]\ar[r,hookrightarrow] & \mathbb{P}_S(E^\vee) \ar[d]\ar[dr] \\ S^{[2]}\ar[r,hookrightarrow] & \Gamma\ar[r,hookrightarrow] & \mathbb{P}(H^0(E)) \end{tikzcd}\tag{14}\] where:

• \(\mathbb{P}_S(E^\vee)\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow}\mathbb{P}(H^0(E))\) is the natural morphism defined by 13 . Its image equals \(\Gamma\), as 13 identifies canonically \(H^0(E\otimes\mathcal{I}_p)\) with the fiber of \(E^\vee\) at \(p\in S\).

• The left square is cartesian, namely \(B\) is the universal length-\(2\) subscheme of 4 .

Observe that \(\mathbb{P}_S(E^\vee)\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow}\mathbb{P}(H^0(E))\) is birational to its image (defines an isomorphism outside \(B\)) and \(\Gamma\) is non-normal, singular along \(S^{[2]}\). Furthermore, there is a unique quadric hypersurface \(Q\subset \mathbb{P}(H^0(E))\) containing \(\Gamma\), defined by the bilinear form \(q\).⁶

Note that by using the Riemann–Roch polynomial, one can compute that \[h^0(S^{[2]},\mathcal{O}(2))=55=h^0(\mathbb{P}^9,\mathcal{O}(2)).\] Therefore, the fact that \(S^{[2]}\) lies on the quadric \(Q\) implies that \(S^{[2]}\) is not projectively normal.

4.4 Degeneracy locus↩︎

The Hilbert square \(S^{[2]}\) can be realized as a degeneracy locus in the smooth quadric \(Q\subset \mathbb{P}(H^0(E))\). This will allow us to describe its ideal in detail in the next subsection.

First let us recall the construction of the spinor bundles on even dimensional quadrics. We refer to [45] for details. Let \(Q\) be a smooth quadric hypersurface of dimension \(2k\), and let \(\mathop{\mathrm{OGr}}_+\mathrel{\vcenter{:}}=\mathop{\mathrm{OGr}}(k+1, 2k+2)_+\) be one of the two corresponding orthogonal Grassmannians, embedded in \(\mathbb{P}^{2^k-1}=\mathbb{P}V_+\) via the spinor embedding. Consider the incidence variety \[\label{eq:incidence-variety} \begin{tikzcd} &I\ar[ld, "p"']\ar[rd, "q"]\mathrel{\vcenter{:}}={\left\{{(x,[V])}:{x\in \mathbb{P}(V)}\right\}}\\ Q&&\mathop{\mathrm{OGr}}_+ \end{tikzcd}\tag{15}\] Here, the map \(q\) is simply the projectivization of the rank-\((k+1)\) universal subbundle on \(\mathop{\mathrm{OGr}}_+\). For the map \(p\), each fiber \(p^{-1}(x)\) is the subvariety \({\left\{{[V]\in \mathop{\mathrm{OGr}}_+}:{x\in \mathbb{P}(V)}\right\}}\), which spans a linear subspace of affine dimension \(2^{k-1}\). This globalizes to the spinor bundle \(\mathcal{S}_+\) on \(Q\), a vector bundle of rank \(2^{k-1}\). We thus view \(\mathcal{S}_+\) as a natural subbundle of the trivial vector bundle \(V_+\otimes\mathcal{O}_Q\), where \(V_+\mathrel{\vcenter{:}}= H^0(\mathop{\mathrm{OGr}}_+, \mathcal{O}(1))^\vee\) is the space of spinors.

Going back to our case where \(Q\) is an \(8\)-dimensional quadric. We note that the fibers \(p^{-1}(x)\) are isomorphic to the previous orthogonal Grassmannian \(\mathop{\mathrm{OGr}}(4,8)\), which in turn is isomorphic to the \(6\)-dimensional quadric. In the above picture of the incidence variety, these \(6\)-dimensional quadrics each span a \(\mathbb{P}^7\) that globalize to the rank-\(8\) spinor bundle \(\mathcal{S}_+\).

On the other hand, recall that the \(K3\) surface \(S\) is determined by the choice of the \(8\)-dimensional linear subspace \[H^0(L)^\vee\subset V_+.\] Equivalently, we get a quotient map \(V_+\mathchoice{\relbar\joinrel\twoheadrightarrow}{\twoheadrightarrow}{\twoheadrightarrow}{\twoheadrightarrow}W\) where \(W\) is also of dimension \(8\). Via composition, we therefore obtain a morphism between two vector bundles of rank \(8\) \[\varphi\colon \mathcal{S}_+\mathchoice{\lhook\joinrel\longrightarrow}{\hookrightarrow}{\hookrightarrow}{\hookrightarrow}V_+\otimes\mathcal{O}_Q \mathchoice{\relbar\joinrel\twoheadrightarrow}{\twoheadrightarrow}{\twoheadrightarrow}{\twoheadrightarrow}W\otimes\mathcal{O}_Q.\]

Thus, when we consider the restriction of the map \(\varphi\) to \(S^{[2]}\), we obtain two vector bundles of rank \(2\), namely, the kernel bundle \(\mathcal{E}_1\mathrel{\vcenter{:}}=\ker(\varphi|_{S^{[2]}})\) and the cokernel bundle \(\mathcal{E}_2\mathrel{\vcenter{:}}=\mathop{\mathrm{coker}}(\varphi|_{S^{[2]}})\).

From the geometric interpretation of the map \(\varphi\), the projectivized kernel bundle \(\mathbb{P}(\mathcal{E}_1)\) is the universal secant line to the \(K3\) surface \(S\). It follows that \(\mathcal{E}_1\) equals the pullback to \(S^{[2]}\) of the universal subbundle on \(\mathop{\mathrm{Gr}}(2,H^0(L)^\vee)\) under the natural map \(S^{[2]}\mathchoice{\lhook\joinrel\longrightarrow}{\hookrightarrow}{\hookrightarrow}{\hookrightarrow}\mathop{\mathrm{Gr}}(2,H^0(L)^\vee)\). In other words, \(\mathcal{E}_1^\vee\) is the tautological rank-\(2\) bundle \(L^{[2]}\) induced by \(L\) and it has first Chern class \(L_2-2\delta\). Thus the rank-\(2\) cokernel bundle \(\mathcal{E}_2\) has first Chern class \(3L_2-7\delta\).

4.5 (Failure of) projective normality↩︎

Since \(S^{[2]}\) can be realized as a degeneracy locus, we can study its ideal using the general machinery of determinantal ideals. Notably, we have the Lascoux resolution which provides a minimal free resolution. In the particular case of the ideal generated by \((n-1)\)-minors of an \(n\times n\)-matrix, the Lascoux resolution specializes to the Gulliksen–Negård complex ([30], see also [58]).

We recall the following properties of spinor bundles [45].

Combining [prop:spinor-properties-1] and [prop:spinor-properties-4] and applying Serre duality, we have for all \(d\ge 1-n\), \(H^n(Q, \mathcal{S}(d))=0\).

Returning to our case of a smooth quadric \(Q\) of dimension \(8\). To simplify notation, we drop the \(+\)-sign and denote the spinor bundle simply by \(\mathcal{S}\). We have \(\det \mathcal{S}\cong\mathcal{O}(-4)\) and \({\mathchoice {\textstyle{\bigwedge\mkern-4.5mu^{7}\mkern 1mu}} {\textstyle{\bigwedge\mkern-4.5mu^{7}\mkern 1mu}} {\scriptstyle{\bigwedge\mkern-5mu^{7}}} {\scriptscriptstyle{\bigwedge\mkern-5mu^{7}}}}\mathcal{S}\cong \mathcal{S}^\vee(-4)\cong \mathcal{S}(-3)\). Following the notation of [2] (that is, the Bourbaki notation), we denote by \(\left\{\alpha_i\right\}_{1\le i\le 5}\) the fundamental weights for the group \(\mathop{\mathrm{SO}}(10)\), and by \(\mathcal{E}_{\alpha_i}\) the corresponding globally generated homogeneous vector bundle of highest weight \(\alpha_i\). Note that \(\mathcal{S}^\vee\) is precisely \(\mathcal{E}_{\alpha_4}\).

Now we prove that the embedding \(S^{[2]}\subset \mathbb{P}^9\) satisfies normality in degree \(d\) for \(d\ge3\). We also show that \(Q\) is the unique quadric hypersurface containing \(S^{[2]}\).

The Gulliksen–Negård complex also allows us to determine the generators for the homogeneous ideal of \(S^{[2]}\) in \(\mathbb{P}^9\).

4.6 The minimal resolution↩︎

Denote by \((X,H)=(S^{[2]},L_2-2\delta)\) the Hilbert square of a very general \(K3\) surface \(S\) of genus \(7\). Our next goal is to deduce a minimal free resolution for \(R_X\mathrel{\vcenter{:}}=\bigoplus_{d\geq0}H^0(X,H^d)\) in terms of the symmetric algebra \(S_X\mathrel{\vcenter{:}}= \mathop{\mathrm{Sym}}^\bullet H^0(X, H)\). Consider the natural map \(m\colon S_X\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow}R_X\) given piece-wise by \[m_d\colon S_d=\mathop{\mathrm{Sym}}^d H^0(X, H)\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow}R_d=H^0(X, H^d).\] We have \(\ker(m_d)=I_d\), the degree-\(d\) piece of the homogeneous ideal of \(X\) in \(\mathbb{P}^9\). From the last section, we know that \(m_d\) is surjective for any \(d\ne 2\). Let \(q\in S_2\) denote one equation for the unique quadric \(Q\mathrel{\vcenter{:}}= Z(q)\) containing \(X\), then we have \(\ker (m_2)=I_2=\mathbb{C}q\), while \(\ker(m_3)=I_3=S_1\cdot q\).

Throughout this section, we will assume that \(X=S^{[2]}\) is exactly the singular locus of the first degeneracy locus \(D\mathrel{\vcenter{:}}= D_7(\varphi)\subset Q\). This is true if \((S,L)\) is chosen to be general among Picard rank one \(K3\) surfaces of genus \(7\), thanks to a Bertini-type argument (see for example [59]). Under this assumption, we are going to prove the following:

Before giving a proof, we need a more detailed analysis of some hypersurfaces in \(\mathbb{P}^9\). Let us consider a quartic polynomial \(f\in S_4\) cutting \(D\) in \(Q\) (recall that \(D=Z(\det(\varphi))\in|\mathcal{O}_Q(4)|\)). The hypersurface \(F\mathrel{\vcenter{:}}= Z(f)\subset\mathbb{P}^9\) may be chosen to be generically smooth along \(X\); indeed, if \(F\) is singular along all of \(X\), it suffices to take \(f+A\cdot q\) for some \(A\in S_2\smallsetminus\langle q\rangle\).

The following lemma describes the intersection of \(X\) with the polar hypersurfaces of \(F\):

In the sequel, we choose homogeneous coordinates \(x_0,...,x_9\) in \(\mathbb{P}^9\) so that \(q=x_0^2+...+x_9^2\). For every \(0\leq i<j\leq 9\), let us consider the quartic polynomial \[f_{ij}\mathrel{\vcenter{:}}=\tfrac{\partial f}{\partial x_i}\cdot x_j-\tfrac{\partial f}{\partial x_j}\cdot x_i\in I_4=\ker(m_4).\] As explained in the proof of 5, up to scalars, these are precisely the \(45\) minors of the Jacobian matrix 19 , hence \(X\) is the intersection in \(\mathbb{P}^9\) of \(Q\), \(F\), and the hypersurfaces \(Z(f_{ij})\). Moreover, we have:

Now we have all the necessary ingredients to prove 3:

4.7 Deformation↩︎

The goal of this section is to show that a general deformation of \(S^{[2]}\) is contained in no quadric of \(\mathbb{P}^9\); consequently, it is projectively normal and we will deduce its entire Betti diagram, which establishes 4.[intro-thmdef-1]. For simplicity, we will denote again by \(X\) the Hilbert square \(S^{[2]}\) and by \(P\) the projective space \(\mathbb{P}^9=\mathbb{P}\left(H^0(L_2-2\delta)^\vee\right)\).

We begin with some cohomological computations of infinitesimal nature.

The following lemma completes the remaining step in the proof of 12.

Now we can derive the desired results for a general deformation of \((S^{[2]},L_2-2\delta)\).

5 Genus 8↩︎

5.1 Quadrics containing the Mukai model↩︎

Let \((S,L)\) be a polarized \(K3\) surface of genus 8, with \(\mathop{\mathrm{Pic}}(S)=\mathbb{Z}\cdot L\). The embedding \(S\mathchoice{\lhook\joinrel\longrightarrow}{\hookrightarrow}{\hookrightarrow}{\hookrightarrow}\mathbb{P}(H^0(L)^\vee)=\mathbb{P}^8\) was described in [2] as follows. Let \(E\) be the unique stable vector bundle with \(v(E)=(2,L,4)\). It is globally generated with \(h^0(E)=6\), and it defines an immersion \[\label{eq:mukai-model8} S\mathchoice{\lhook\joinrel\longrightarrow}{\hookrightarrow}{\hookrightarrow}{\hookrightarrow}\mathop{\mathrm{Gr}}(2,H^0(E)^\vee)\mathchoice{\lhook\joinrel\longrightarrow}{\hookrightarrow}{\hookrightarrow}{\hookrightarrow}\mathbb{P}({\mathchoice {\textstyle{\bigwedge\mkern-4.5mu^{2}\mkern 1mu}} {\textstyle{\bigwedge\mkern-4.5mu^{2}\mkern 1mu}} {\scriptstyle{\bigwedge\mkern-5mu^{2}}} {\scriptscriptstyle{\bigwedge\mkern-5mu^{2}}}} H^0(E)^\vee)\tag{24}\] such that \(S\) is the intersection of \(\mathop{\mathrm{Gr}}(2,H^0(E)^\vee)\) with the subspace \(\mathbb{P}(H^0(L)^\vee)\subset \mathbb{P}({\mathchoice {\textstyle{\bigwedge\mkern-4.5mu^{2}\mkern 1mu}} {\textstyle{\bigwedge\mkern-4.5mu^{2}\mkern 1mu}} {\scriptstyle{\bigwedge\mkern-5mu^{2}}} {\scriptscriptstyle{\bigwedge\mkern-5mu^{2}}}} H^0(E)^\vee)\) (this inclusion being defined by the determinant map \({\mathchoice {\textstyle{\bigwedge\mkern-4.5mu^{2}\mkern 1mu}} {\textstyle{\bigwedge\mkern-4.5mu^{2}\mkern 1mu}} {\scriptstyle{\bigwedge\mkern-5mu^{2}}} {\scriptscriptstyle{\bigwedge\mkern-5mu^{2}}}} H^0(E)\mathchoice{\relbar\joinrel\twoheadrightarrow}{\twoheadrightarrow}{\twoheadrightarrow}{\twoheadrightarrow}H^0(L)\)).

The vector space of quadrics \(I_S(2)\) equals the vector space of quadrics in \(\mathbb{P}({\mathchoice {\textstyle{\bigwedge\mkern-4.5mu^{2}\mkern 1mu}} {\textstyle{\bigwedge\mkern-4.5mu^{2}\mkern 1mu}} {\scriptstyle{\bigwedge\mkern-5mu^{2}}} {\scriptscriptstyle{\bigwedge\mkern-5mu^{2}}}} H^0(E)^\vee)\) containing \(\mathop{\mathrm{Gr}}(2,H^0(E)^\vee)\); this is canonically identified with \({\mathchoice {\textstyle{\bigwedge\mkern-4.5mu^{2}\mkern 1mu}} {\textstyle{\bigwedge\mkern-4.5mu^{2}\mkern 1mu}} {\scriptstyle{\bigwedge\mkern-5mu^{2}}} {\scriptscriptstyle{\bigwedge\mkern-5mu^{2}}}} H^0(E)^\vee\) (any element in \({\mathchoice {\textstyle{\bigwedge\mkern-4.5mu^{2}\mkern 1mu}} {\textstyle{\bigwedge\mkern-4.5mu^{2}\mkern 1mu}} {\scriptstyle{\bigwedge\mkern-5mu^{2}}} {\scriptscriptstyle{\bigwedge\mkern-5mu^{2}}}} H^0(E)^\vee\) defines by wedge a symmetric bilinear form on \({\mathchoice {\textstyle{\bigwedge\mkern-4.5mu^{2}\mkern 1mu}} {\textstyle{\bigwedge\mkern-4.5mu^{2}\mkern 1mu}} {\scriptstyle{\bigwedge\mkern-5mu^{2}}} {\scriptscriptstyle{\bigwedge\mkern-5mu^{2}}}} H^0(E)^\vee\)). We obtain \[I_S(2)\cong {\mathchoice {\textstyle{\bigwedge\mkern-4.5mu^{2}\mkern 1mu}} {\textstyle{\bigwedge\mkern-4.5mu^{2}\mkern 1mu}} {\scriptstyle{\bigwedge\mkern-5mu^{2}}} {\scriptscriptstyle{\bigwedge\mkern-5mu^{2}}}} H^0(E)^\vee,\] so that \(\mathop{\mathrm{Gr}}(4,H^0(E))=\mathop{\mathrm{Gr}}(2,H^0(E)^\vee)\subset \mathbb{P}({\mathchoice {\textstyle{\bigwedge\mkern-4.5mu^{2}\mkern 1mu}} {\textstyle{\bigwedge\mkern-4.5mu^{2}\mkern 1mu}} {\scriptstyle{\bigwedge\mkern-5mu^{2}}} {\scriptscriptstyle{\bigwedge\mkern-5mu^{2}}}} H^0(E)^\vee)\) is naturally identified with \[\psi_2:\mathop{\mathrm{Gr}}(4,H^0(E))\overset{\cong}{\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow}}Y_2 \mathchoice{\lhook\joinrel\longrightarrow}{\hookrightarrow}{\hookrightarrow}{\hookrightarrow}|I_S(2)|.\] (recall 7 ). In this way, 24 can be regarded as an immersion of \(S\) into the space of quadrics containing it, admitting the following geometric interpretation.

As we proved in 3 (see 4), the complete linear system \(|L_2-2\delta|\) on \(S^{[2]}\) defines a closed immersion \[S^{[2]}\mathchoice{\lhook\joinrel\longrightarrow}{\hookrightarrow}{\hookrightarrow}{\hookrightarrow}|I_S(2)|^\vee=\mathbb{P}({\mathchoice {\textstyle{\bigwedge\mkern-4.5mu^{2}\mkern 1mu}} {\textstyle{\bigwedge\mkern-4.5mu^{2}\mkern 1mu}} {\scriptstyle{\bigwedge\mkern-5mu^{2}}} {\scriptscriptstyle{\bigwedge\mkern-5mu^{2}}}} H^0(E))\] which factors through \(\mathop{\mathrm{Gr}}(2,H^0(E))\), via the natural map \(\xi\mathchoice{\longmapsto}{\mapsto}{\mapsto}{\mapsto}H^0(E\otimes\mathcal{I}_\xi)\). Thus we can consider the stratification \[S^{[2]}\mathchoice{\lhook\joinrel\longrightarrow}{\hookrightarrow}{\hookrightarrow}{\hookrightarrow}\Gamma \mathchoice{\lhook\joinrel\longrightarrow}{\hookrightarrow}{\hookrightarrow}{\hookrightarrow}\mathop{\mathrm{Gr}}(2,H^0(E)),\] where \(\Gamma\) is the locus of \(V\in\mathop{\mathrm{Gr}}(2,H^0(E))\) such that \(V\subset H^0(E\otimes\mathcal{I}_p)\) for some \(p\in S\). Similarly to the case of genus \(7\) (see 14 ), there is a commutative diagram \[\begin{tikzcd} B \ar[d,"\pi"]\ar[r,hookrightarrow] & \mathop{\mathrm{Gr}}_S(2,K) \ar[d]\ar[dr] \\ S^{[2]}\ar[r,hookrightarrow] & \Gamma\ar[r,hookrightarrow] & \mathop{\mathrm{Gr}}(2,H^0(E)) \end{tikzcd}\] where:

• \(K\) is the kernel bundle on \(S\) sitting in the short exact sequence \[0\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow}K\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow}H^0(E)\otimes\mathcal{O}_S\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow}E\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow}0.\] In other words, \(K^\vee\) is the restriction to \(S\) of the rank-4 tautological quotient bundle on \(\mathop{\mathrm{Gr}}(2,H^0(E)^\vee)\).

• \(\mathop{\mathrm{Gr}}_S(2,K)\) is the Grassmannian bundle defined by \(K\), and \(\mathop{\mathrm{Gr}}_S(2,K)\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow}\mathop{\mathrm{Gr}}(2,H^0(E))\) is the natural morphism. Its image equals \(\Gamma\).

• The left square is cartesian, namely \(B\) is the universal length-\(2\) subscheme of 4 .

• The morphism \(\mathop{\mathrm{Gr}}_S(2,K)\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow}\Gamma\) is birational (defines an isomorphism outside \(B\)) and \(\Gamma\) is non-normal, singular along \(S^{[2]}\).

5.2 Degeneracy locus↩︎

The Hilbert square \(S^{[2]}\) can again be realized as a degeneracy locus, in a strikingly similar fashion to the case of genus \(7\). In the sequel, we will denote by \(G\) the Grassmannian \(G\mathrel{\vcenter{:}}=\mathop{\mathrm{Gr}}(2,H^0(E))\). Consider the incidence variety \[\begin{tikzcd} &I\ar[ld, "p"']\ar[rd, "q"]\mathrel{\vcenter{:}}={\left\{{([U],[V])}:{(U,V)=0}\right\}}\\ G\mathrel{\vcenter{:}}=\mathop{\mathrm{Gr}}(2,H^0(E))&&\mathop{\mathrm{Gr}}(2,H^0(E)^\vee) \end{tikzcd}\] Note that \(I\) is nothing but the flag variety \(\mathop{\mathrm{Fl}}(2,4,H^0(E))\). Each fiber of \(p\) is a 4-dimensional quadric in \(\mathop{\mathrm{Gr}}(2,H^0(E)^\vee)\), spanning a \(\mathbb{P}^5\) in \(\mathbb{P}({\mathchoice {\textstyle{\bigwedge\mkern-4.5mu^{2}\mkern 1mu}} {\textstyle{\bigwedge\mkern-4.5mu^{2}\mkern 1mu}} {\scriptstyle{\bigwedge\mkern-5mu^{2}}} {\scriptscriptstyle{\bigwedge\mkern-5mu^{2}}}} H^0(E)^\vee)\). This construction globalizes to the rank-6 vector bundle \({\mathchoice {\textstyle{\bigwedge\mkern-4.5mu^{2}\mkern 1mu}} {\textstyle{\bigwedge\mkern-4.5mu^{2}\mkern 1mu}} {\scriptstyle{\bigwedge\mkern-5mu^{2}}} {\scriptscriptstyle{\bigwedge\mkern-5mu^{2}}}}\mathcal{Q}^\vee\) (where \(\mathcal{Q}\) is the rank-4 tautological quotient on \(G\)), which is naturally a subbundle of the trivial vector bundle \({\mathchoice {\textstyle{\bigwedge\mkern-4.5mu^{2}\mkern 1mu}} {\textstyle{\bigwedge\mkern-4.5mu^{2}\mkern 1mu}} {\scriptstyle{\bigwedge\mkern-5mu^{2}}} {\scriptscriptstyle{\bigwedge\mkern-5mu^{2}}}} H^0(E)^\vee\otimes\mathcal{O}_G\). We denote it by \(M\mathrel{\vcenter{:}}= {\mathchoice {\textstyle{\bigwedge\mkern-4.5mu^{2}\mkern 1mu}} {\textstyle{\bigwedge\mkern-4.5mu^{2}\mkern 1mu}} {\scriptstyle{\bigwedge\mkern-5mu^{2}}} {\scriptscriptstyle{\bigwedge\mkern-5mu^{2}}}}\mathcal{Q}^\vee\) from now on. Note that there exists a natural pairing \({\mathchoice {\textstyle{\bigwedge\mkern-4.5mu^{2}\mkern 1mu}} {\textstyle{\bigwedge\mkern-4.5mu^{2}\mkern 1mu}} {\scriptstyle{\bigwedge\mkern-5mu^{2}}} {\scriptscriptstyle{\bigwedge\mkern-5mu^{2}}}}\mathcal{Q}\otimes{\mathchoice {\textstyle{\bigwedge\mkern-4.5mu^{2}\mkern 1mu}} {\textstyle{\bigwedge\mkern-4.5mu^{2}\mkern 1mu}} {\scriptstyle{\bigwedge\mkern-5mu^{2}}} {\scriptscriptstyle{\bigwedge\mkern-5mu^{2}}}}\mathcal{Q}\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow}{\mathchoice {\textstyle{\bigwedge\mkern-4.5mu^{4}\mkern 1mu}} {\textstyle{\bigwedge\mkern-4.5mu^{4}\mkern 1mu}} {\scriptstyle{\bigwedge\mkern-5mu^{4}}} {\scriptscriptstyle{\bigwedge\mkern-5mu^{4}}}}\mathcal{Q}\cong \mathcal{O}_G(1)\), and so \(M^\vee\cong M(1)\).

On the other hand, the \(K3\) surface \(S\) is determined by the choice of the \(9\)-dimensional linear subspace \[H^0(L)^\vee\subset {\mathchoice {\textstyle{\bigwedge\mkern-4.5mu^{2}\mkern 1mu}} {\textstyle{\bigwedge\mkern-4.5mu^{2}\mkern 1mu}} {\scriptstyle{\bigwedge\mkern-5mu^{2}}} {\scriptscriptstyle{\bigwedge\mkern-5mu^{2}}}} H^0(E)^\vee,\] so we get a quotient map \({\mathchoice {\textstyle{\bigwedge\mkern-4.5mu^{2}\mkern 1mu}} {\textstyle{\bigwedge\mkern-4.5mu^{2}\mkern 1mu}} {\scriptstyle{\bigwedge\mkern-5mu^{2}}} {\scriptscriptstyle{\bigwedge\mkern-5mu^{2}}}} H^0(E)^\vee\mathchoice{\relbar\joinrel\twoheadrightarrow}{\twoheadrightarrow}{\twoheadrightarrow}{\twoheadrightarrow}W\) where \(W\) is of dimension \(6\). Via composition, we obtain a morphism between two vector bundles of rank \(6\) \[\varphi\colon M \mathchoice{\lhook\joinrel\longrightarrow}{\hookrightarrow}{\hookrightarrow}{\hookrightarrow}{\mathchoice {\textstyle{\bigwedge\mkern-4.5mu^{2}\mkern 1mu}} {\textstyle{\bigwedge\mkern-4.5mu^{2}\mkern 1mu}} {\scriptstyle{\bigwedge\mkern-5mu^{2}}} {\scriptscriptstyle{\bigwedge\mkern-5mu^{2}}}} H^0(E)^\vee\otimes \mathcal{O}_G\mathchoice{\relbar\joinrel\twoheadrightarrow}{\twoheadrightarrow}{\twoheadrightarrow}{\twoheadrightarrow}W\otimes\mathcal{O}_G.\] The same argument as in 6 applies, mutatis mutandis, to the following result:

5.3 Projective normality↩︎

This section is concerned with the equations of \(S^{[2]}\) in \(\mathbb{P}^{14}=\mathbb{P}\left({\mathchoice {\textstyle{\bigwedge\mkern-4.5mu^{2}\mkern 1mu}} {\textstyle{\bigwedge\mkern-4.5mu^{2}\mkern 1mu}} {\scriptstyle{\bigwedge\mkern-5mu^{2}}} {\scriptscriptstyle{\bigwedge\mkern-5mu^{2}}}} H^0(E)\right)\). Keeping in mind that \(G=\mathop{\mathrm{Gr}}(2, H^0(E))\) is projectively normal, to establish the projective normality of \(S^{[2]}\subset\mathbb{P}^{14}\) it is sufficient to show the vanishing of \(H^1(G, \mathcal{I}_{S^{[2]}/G}(d))\) for all \(d\ge2\).

Let us finish this section with some comments on 4.[intro-thmdef-2], concerning a general deformation \((X,H)\in \mathcal{M}_6^{(1)}\) of \(S^{[2]}\) in \(\mathbb{P}^{14}\). First note that \(X\) is indeed projectively normal, as \(S^{[2]}\) itself is. Using the Riemann–Roch polynomial, we have \(h^0(\mathbb{P}^{14}, \mathcal{I}_{X/\mathbb{P}^{14}}(2))=15\) and \(h^0(\mathbb{P}^{14}, \mathcal{I}_{X/\mathbb{P}^{14}}(3))=245\), and so the image of the map \[H^0(\mathbb{P}^{14}, \mathcal{I}_{X/\mathbb{P}^{14}}(2))\otimes H^0(\mathbb{P}^{14}, \mathcal{O}(1))\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow}H^0(\mathbb{P}^{14}, \mathcal{I}_{X/\mathbb{P}^{14}}(3))\] has codimension at least \(20\). Therefore, to generate the homogeneous ideal of \(X\) we need \(15\) quadrics and at least \(20\) cubics—note that in the case of \(S^{[2]}\), we needed \(55\) cubics due to the \(35\) linear syzygies among the Plücker quadrics.

We prove in the forthcoming work [33] that a general \((X,H)\in \mathcal{M}_6^{(1)}\) is the singular locus of a unique cubic hypersurface in \(\mathbb{P}^{14}\). In other words, similarly to the case of square \(4\), the \(15\) quadrics are exactly the polars of this cubic, and \(X\) will be the scheme-theoretical intersection of them. The extra cubics are needed to generate the saturated homogeneous ideal.

This would be quite different from the varieties of lines for cubic fourfolds, which have the same square \(6\) but divisibility \(2\): for these, the \(15\) quadrics are always the Plücker quadrics, so they are not sufficient at all to recover the variety.

6 Final questions and comments↩︎

Apart from the aforementioned work [33] on Coble type hypersurfaces, our study naturally raises several questions pointing towards a more systematic understanding of linear systems on hyperkähler fourfolds of \(K3^{[2]}\)-type.

In the case of polarized Hilbert squares \((S^{[2]},L_2-2\delta)\), our good understanding of the projective geometry and equations in genus 7 and 8 strongly relies on the existence of a Mukai model for \(S\). In particular, we do not have a very satisfactory geometric realization of the multiplication maps \[\mathop{\mathrm{Sym}}^d H^0(S^{[2]},L_2-2\delta)\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow}H^0\left(S^{[2]},(L_2-2\delta)^{\otimes d}\right)\] in arbitrary genus. Nevertheless, in view of our results (as well as some supporting computer experiments), it is reasonable to expect:

For general polarized hyperkähler fourfolds of \(K3^{[2]}\)-type that are not actual Hilbert squares, the bounds in 2 are no longer optimal. Namely, we have showed that projective normality holds also in the case of square \(4\), which is indeed predicted by the Hilbert function; similarly, an elementary computation shows that generation of the homogeneous ideal by quadrics may also be satisfied by hyperkähler fourfolds of \(K3^{[2]}\)-type equipped with a polarization of square \(8\). In much greater generality, it is possible to compute a theoretical Betti diagram by imposing that the syzygies of a hyperkähler fourfold \((X,H)\in\mathcal{M}_{2d}^{(\gamma)}\) are as simple as possible (subject to some relations such as Green’s duality 1). The reader may consult [62] for further details, including an interactive SageMathCell where the theoretical Betti diagrams in low degrees are computed.

It is thus natural to pose the following question:

Note that our 4.[intro-thmdef-1] settles this question in the affirmative for \(2d=4\). On the other hand, the same question will not work verbatim in higher divisibility: for a general element of \((X,H)\in\mathcal{M}_6^{(2)}\), the variety of lines of a cubic fourfold, the 15 quadrics containing \(X\)—namely the 15 Plücker quadrics containing \(\mathop{\mathrm{Gr}}(2,6)\)—have unexpected linear relations.

Finally, it would also be interesting to understand very ampleness in higher dimensions. Following [24], let us denote by \({}^{m}\!{\mathcal{M}}_{2d}^{(\gamma)}\) the moduli space of hyperkähler manifolds of \(K3^{[m]}\)-type equipped with a polarization of square \(2d\) and divisibility \(\gamma\). In analogy with the cases \(m=1\) and \(m=2\), it is natural to expect:

To the best of our knowledge, very ampleness is only known in the range \(d\geq m+1\) (see [24]). Note that the analogous question may have a negative answer for higher divisibility: as shown in [15], for a general \((X,H)\in{}^{3}\!{\mathcal{M}}_{4}^{(2)}\) the corresponding morphism \(X\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow}\mathbb{P}\left(H^0(X,H)^\vee\right)=\mathbb{P}^{19}\) has degree 2 onto its image.

References↩︎

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1. Note that the space \(\mathcal{M}_{2d}^{(\gamma)}\) is non-empty and irreducible of dimension \(20\) if and only if \(\gamma=1\), or \(\gamma=2\) and \(2d\equiv 6\pmod 8\) (see 2.1). In consequence, when \(2d\not\equiv 6\pmod 8\), we will simply speak of a polarization of square \(2d\) without specifying the divisibility, which is always \(1\).↩︎

2. Note that the extra equations of higher degree are needed to generate the (saturated) homogeneous ideal.↩︎

3. In the case where \(X\) is projectively normal, \(R_X\) is simply the quotient \(S_X/I_X\) and is usually referred to as the coordinate ring of \(X\). We adopt the terminology to distinguish the two in cases where \(X\) is not projectively normal. Another appropriate name would be the ring of power.↩︎

4. By the assumption that \(\omega_X\cong \mathcal{O}_X\), this is equivalent to \(h^1(\mathcal{O}_X)=0\).↩︎

5. Using the fact that \(Y_0\) admits a (rational) two-to-one covering by the moduli space \(\mathcal{M}(2,L,2)\) via \(\psi_0\).↩︎

6. More precisely, the map \(E^\vee\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow}H^0(E)\otimes\mathcal{O}_S\) in [bilformgenus7] identifies, via the natural isomorphism \(H^0(E)\cong H^0(E)^\vee\), with the map \(E^\vee\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow}H^0(E)^\vee\otimes \mathcal{O}_S\) obtained by dualizing [bilformgenus7]. Therefore, quadrics containing \(\Gamma\) are parametrized by the kernel of \(\mathop{\mathrm{Sym}}^2 H^0(E)\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow}H^0(\mathop{\mathrm{Sym}}^2 E)\).↩︎

7. The four terms that are marked with an underline contain extra syzygies not expected from computing with the Hilbert function. Indeed, we will see in ?? that these do not appear for a general deformation of \(S^{[2]}\).↩︎

8. This is a pairing \(\mathop{\mathrm{Sym}}^2\mathcal{T}_{Q}\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow}\mathcal{O}_Q(2)\), fiberwise given by restriction of \(Q\) to its Zariski tangent space.↩︎

9. This follows, for instance, from the sheaf cohomology interpretation of \(\mathcal{K}_{i,j}(X',H')\) for every \(j\geq1\) (see [61]).↩︎