Hodge Structures of K3 type of bidouble covers of rational surfaces

TL;DR

This paper studies the Hodge structures of bidouble covers of rational surfaces, classifies certain cases with specific geometric properties, and explores related conjectures and new constructions for surfaces with higher geometric genus.

Contribution

It classifies bidouble covers over rational surfaces with specific intermediate quotients, analyzes their Hodge structures, and introduces iterated covers to produce surfaces with higher geometric genus.

Findings

Classified all minimal bidouble covers with $p_g=1,2,3$.

Analyzed the Infinitesimal Torelli Property and related conjectures for these surfaces.

Introduced iterated bidouble covers to construct surfaces with higher $p_g$ while controlling Hodge structures.

Abstract

A bidouble cover is a flat $G:=\left(\mathbb{Z}/2\mathbb{Z}\right)^2$-Galois cover $X \rightarrow Y$. In this situation there exist three intermediate quotients $Y_1,Y_2$ and $Y_3$ which correspond to the three subgroups $\mathbb{Z}/2\mathbb{Z} \leq G$. In this paper we consider the following situation: $Y$ will be a rational surface and $Y_i$ will be either a surface with $p_g=0$ or a K3 surface. These assumptions will enable us to have a strong control on the weight 2 Hodge structure of the covering surface $X$. In particular, we classify all covers with these properties if $Y$ is minimal, obtaining surfaces $X$ with $p_g(X)=1,2,3$. Moreover, we will discuss the Infinitesimal Torelli Property, the Chow groups and Chow motive, and the Tate and Mumford-Tate conjectures for $X$. We also introduce another construction, called iterated bidouble cover, which allows us to obtain surfaces with higher value of $p_g$ for which we still have a strong control on the weight 2 Hodge structure.