Deformations and moduli of irregular canonical covers with $K^2=4p_g-8$

TL;DR

This paper investigates the deformation theory and moduli spaces of irregular surfaces of general type with specific canonical invariants, revealing that most deformations are two-to-one covers of ruled surfaces and establishing the existence of infinitely many moduli spaces with complex structure.

Contribution

It classifies deformations of canonical Galois covers of degree four, showing they are mostly non-birational and unobstructed, and demonstrates the existence of infinitely many moduli spaces with jumping canonical degrees.

Findings

Most deformations factor through double covers of ruled surfaces.

General deformations are typically two-to-one onto their images.

The moduli space components are uniruled and contain proper quadruple subloci.

Abstract

In this article, we study the moduli of irregular surfaces of general type with at worst canonical singularities satisfying $K_X^2 = 4p_g(X)-8$, for any even integer $p_g\geq 4$. These surfaces also have unbounded irregularity $q$. We carry out our study by investigating the deformations of the canonical morphism $\varphi:X\to \mathbb{P}^N$, where $\varphi$ is Galois of degree 4. These canonical covers are classified in by the first two authors into four distinct families. We show that any deformation of $\varphi$ factors through a double cover of a ruled surface and, hence, is never birational. More interestingly, we prove that, with two exceptions, a general deformation of $\varphi$ is two-to-one onto its image, whose normalization is a ruled surface of appropriate irregularity. We also show that with the exception of one family, the deformations of $X$ are unobstructed, and consequently, $X$ belongs to a unique irreducible component of the Gieseker moduli space, which we prove is uniruled. As a result of all this, we show the existence of infinitely many moduli spaces, satisfying the strict Beauville inequality $p_g > 2q-4$, with an irreducible component that has a proper "quadruple" sublocus where the degree of the canonical morphism jumps up. The existence of jumping subloci is a contrast with the moduli of surfaces with $K_X^2 = 2p_g - 4$, studied by Horikawa. There is a similarity and difference to the moduli of curves of genus $g\geq 3$, for, like in the case of curves, the degree of the canonical morphism goes down outside a closed sublocus but, unlike in the case of curves, it is never birational.