Calabi-Yau threefolds and moduli of Abelian surfaces. I. (Q2743770)

The paper describes birational models of the moduli spaces \({\mathcal A}_d\) of polarized abelian surfaces of type \((1, d)\) and their coverings \({\mathcal A}^{\text{lev}}_d\) with canonical level structure, for some small numbers \(d\). It is shown in particular that \({\mathcal A}^{\text{lev}}_6\) is birational to a nonsingular quadric hypersurface in \(\mathbb{P}^4\), \({\mathcal A}^{\text{lev}}_8\) is birational to a rational conic bundle over \(\mathbb{P}^2\), and \({\mathcal A}^{\text{lev}}_{10}\) is birational to a singular prime Fano 3-fold of genus 9, index 1 in \(\mathbb{P}^{10}\). In all cases the abelian surfaces are related to some Calabi-Yau 3-folds. Here I will point out only the case \(d= 8\): If \(A\subset\mathbb{P}^7\) is a general abelian surface of type \((1, 8)\), its homogeneous ideal is generated by 4 quadrics and 16 cubics. The 4 quadrics cut out a complete intersection \(X\) in \(\mathbb{P}^7\), whose small resolution is a Calabi Yau 3-fold. \(X\) contains a pencil of \((1,8)\)-polarized abelian surfaces, and this gives a description of \({\mathcal A}^{\text{lev}}_9\) as a \(\mathbb{P}^1\)-bundle over an open subset of \(\mathbb{P}^2\). Some of the Calabi-Yau 3-folds were not known before. There is a nice analogy betweeen the Calabi-Yau's described in the paper and the del Pezzo surfaces which are double covers of \(\mathbb{P}^2\).