Lagrangian 4-planes in holomorphic symplectic varieties of \(K3^{[4]}\)-type. (English) Zbl 1307.14014
Let \(S\) be a \(K3\) surface and \((\cdot,\cdot)\) be the intersection form on \(S\). It is known that one can characterize the Mori cone of effective curves of \(S\) by means of nodal classes, i.e. effective curve classes \(C\) for which \((C,C)=-2\).
With this in mind, B. Hassett and Y. Tschinkel [Asian J. Math. 14, No. 3, 303–322 (2010; Zbl 1216.14012)] conjectured an analogue for irreducible holomorphic symplectic varieties, i.e. they conjecture that the Mori cone of an irreducible holomorphic symplectic variety is controlled by nodal classes \(C\) such that \((C,C)=-\gamma\) where the \((\cdot, \cdot)\) is the Beauville-Bogomolov form and \(\gamma\) is related to the extremal contraction associated to \(C\).
More precisely, when \(X\) is deformation-equivalent to the Hilbert scheme of \(n\) points on a \(K3\) surface \(S\), the form of the conjecture becomes:
Conjecture (Hasset, Tschinkel). Let \(X\) be of \(K3^{[n]}\)-type, let \(\mathbb P^n \subset X\) a smoothly embedded Lagrangian \(n\)-plane, and \(\ell\in H^2(X;\mathbb Z)\) the class of the line in \(\mathbb P^n\). Then \((\ell,\ell) = -\frac{n+3}{2}\).
The conjecture has been proven for \(n=2,3\) and the paper provides the proof for \(n=4\). The proof uses representation theory of the monodromy group of \(X\), in order to relate the intersection theory of \(X\) to that of a Hilbert scheme of 4 points on a \(K3\) surface. This also allows the authors to determine the class of the Lagrangian 4-plane classes on \(X\) of \(K3^{[4]}\)-type and show that there is a unique monodromy orbit of Lagrangian 4-planes.
With this in mind, B. Hassett and Y. Tschinkel [Asian J. Math. 14, No. 3, 303–322 (2010; Zbl 1216.14012)] conjectured an analogue for irreducible holomorphic symplectic varieties, i.e. they conjecture that the Mori cone of an irreducible holomorphic symplectic variety is controlled by nodal classes \(C\) such that \((C,C)=-\gamma\) where the \((\cdot, \cdot)\) is the Beauville-Bogomolov form and \(\gamma\) is related to the extremal contraction associated to \(C\).
More precisely, when \(X\) is deformation-equivalent to the Hilbert scheme of \(n\) points on a \(K3\) surface \(S\), the form of the conjecture becomes:
Conjecture (Hasset, Tschinkel). Let \(X\) be of \(K3^{[n]}\)-type, let \(\mathbb P^n \subset X\) a smoothly embedded Lagrangian \(n\)-plane, and \(\ell\in H^2(X;\mathbb Z)\) the class of the line in \(\mathbb P^n\). Then \((\ell,\ell) = -\frac{n+3}{2}\).
The conjecture has been proven for \(n=2,3\) and the paper provides the proof for \(n=4\). The proof uses representation theory of the monodromy group of \(X\), in order to relate the intersection theory of \(X\) to that of a Hilbert scheme of 4 points on a \(K3\) surface. This also allows the authors to determine the class of the Lagrangian 4-plane classes on \(X\) of \(K3^{[4]}\)-type and show that there is a unique monodromy orbit of Lagrangian 4-planes.
Reviewer: Paola Comparin (Consepción)
Citations:
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