Prym varieties and the canonical map of surfaces of general type

TL;DR

This paper explores the construction of infinite series of surfaces of general type with canonical maps of degree two onto embedded surfaces, using Prym varieties and a new concept called good generating pairs.

Contribution

It introduces the notion of good generating pairs and demonstrates how they generate new infinite series of surfaces with specific canonical map properties, extending Beauville's previous work.

Findings

Constructed new infinite series of surfaces of general type with degree 2 canonical maps.

Proved that good generating pairs have bounded invariants.

Showed that only two examples exist with linear systems of dimension greater than one.

Abstract

Let X be a smooth complex surface of general type such that the image of the canonical map $\phi$ of X is a surface $\Sigma$ and that $\phi$ has degree $\delta\geq 2$. Let $\epsilon\colon S\to \Sigma$ be a desingularization of $\Sigma$ and assume that the geometric genus of S is not zero. Beauville has proved that in this case S is of general type and $\epsilon$ is the canonical map of S. Beauville has also constructed the only infinite series of examples $\phi:X\to \Sigma$ with the above properties that was known up to now. Starting from his construction, we define a {\em good generating pair}, namely a pair $(h:V\to W, L)$ where h is a finite morphism of surfaces and L is a nef and big line bundle of W satisfying certain assumptions. We show that by applying a construction analogous to Beauville's to a good generating pair one obtains an infinite series of surfaces of general type whose canonical map is 2-to-1 onto a canonically embedded surface. In this way we are able to construct more infinite series of such surfaces. In addition, we show that good generating pairs have bounded invariants and that there exist essentially only 2 examples with $\dim |L|>1$. The key fact that we exploit for obtaining these results is that the Albanese variety P of V is a Prym variety and that the fibre of the Prym map over P has positive dimension.