Views on level $\ell$ curves, K3 surfaces and Fano threefolds

Abstract

An analogue of the Mukai map $m_g:{\mathcal{P}}_g\to {\mathcal{M}}_g$ is studied for the moduli ${\mathcal{R}}_{g,\ell }$ of genus $g$ curves $C$ with a level $\ell$ structure. Let ${\mathcal{P}}_{g,\ell }^{\perp }$ be the moduli space of $4$-tuples $(S,\mathcal{L},\mathcal{E},C)$ so that $(S,\mathcal{L})$ is a polarized K3 surface of genus $g$, $\mathcal{E}$ is orthogonal to $\mathcal{L}$ in Pic$S$ and defines a standard degree $\ell$ K3 cyclic cover of $S$, $C\in |\mathcal{L}|$. We say that $(S,\mathcal{L},\mathcal{E})$ is a level $\ell$ K3 surface. These exist for $\ell \le 8$ and their families are known. We define a level $\ell$ Mukai map $r_{g,\ell }:{\mathcal{P}}_{g,\ell }^{\perp }\to {\mathcal{R}}_{g,\ell }$, induced by the assignment of $(S,\mathcal{L},\mathcal{E},C)$ to $(C,\mathcal{E}\otimes {\mathcal{O}}_C)$. We investigate a curious possible analogy between $m_g$ and $r_{g,\ell }$, that is, the failure of the maximal rank of $r_{g,\ell }$ for $g=g_{\ell }\pm 1$, where $g_{\ell }$ is the value of $g$ such that $\dim {\mathcal{P}}_{g,\ell }^{\perp }=\dim {\mathcal{R}}_{g,\ell }$. This is proven here for $\ell =3$. As a related open problem we discuss Fano threefolds whose hyperplane sections are level $\ell$ K3 surfaces and their classification.

19 pages. To be published in the special issue of Bollettino dell'Unione Matematica Italiana dedicated to Fabrizio Catanese

References in corpus (1)

• On K3 surface quotients of K3 or Abelian surfaces