Remark on polarized \(K3\) surfaces of genus 36
A smooth \(K3\) surface is a smooth projective complex surface \(S\) such that \(\omega_S \cong \mathcal{O}_S\) and \(H^1(S,\mathcal{O}_S)=0\). It is well known that the moduli space \(\mathcal{K}_g\) of smooth polarized \(K3\) surfaces \((S,A)\) such that \(A^2=2g-2\) is an irreducible quasiprojective variety. S. Mukai has shown in a series of papers that \(\mathcal{K}_g\) is unirational for \(g \in \{2,\dots,10,12,13,18,20\}\). However \(\mathcal{K}_g\) is not unirational for general \(g \geq 43\). In principle, the proof that \(\mathcal{K}_g\) is unirational is based on the fact that a general \(K3\) surface \(S\) is an anticanonical section of some smooth Fano threefold \(X\). This implies that the map \(\mathcal{F}_g \rightarrow \mathcal{K}_g\) is generically surjective, where \(\mathcal{F}_g\) is the moduli space of pairs \((X_g,S_g)\), where \(X_g\) is a Fano threefold of genus \(g\) and \(S_g \in |-K_{X_g}|\) is smooth. From this the unirationality of \(\mathcal{K}_g\) follows. The previous method raises the fundamental question when is a smooth \(K3\) surface an anticanonical member of a smooth Fano threefold. Suppose that \(S \in |-K_X|\), where \(X\) is a smooth Fano threefold of genus \(g\). Let \(R_X=\mathrm{Pic}(X)\) be the Picard lattice of \(X\) equipped with the pairing \((D_1,D_2)=-D_1\cdot D_2\cdot K_X\). Then there is a primitive embedding \(R_X \rightarrow \mathrm{Pic}(S)\) which maps \(-K_X\) to an ample class in \(S\) of square \(2g-2\). A smooth \(K3\) surface \(S\) with such an embedding is called a \(K3\) surface of type \(R_X\). A. Beauville has shown that the converse is also true, i.e., that every \(K3\) surface \(S\) of type \(R_X\) is an anticanonical member of some smooth Fano threefold \(Y\) [\textit{A. Beauville}, The Fano conference. Papers of the conference organized to commemorate the 50th anniversary of the death of Gino Fano (1871--1952), Torino, Italy, September 29--October 5, 2002. Torino: Università di Torino, Dipartimento di Matematica. 175--184 (2004; Zbl 1096.14034)]. In this paper the author shows that the moduli space \(\mathcal{K}_{36}^R\) of smooth \(K3\) surfaces \(S\) of genus 36 of type \(R\) is unirational, where \(R=\mathbb{Z}^2\) equipped with the quadratic form \(70x^2+4xy-2y^2\). In order to prove this the author generalizes Beauville's proof in order to show that a general \(K3\) surface of type \(R\) is a smooth anticanonical member of a certain Fano threefold \(X\) of genus 36 with canonical Gorenstein singularities.
scientific article; zbMATH DE number 6062592
scientific article