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

Purnaprajna Bangere, Francisco Javier Gallego, Jayan Mukherjee, Debaditya Raychaudhury

Published 2021-08-12Version 1

In this article we study the deformations of the canonical morphism $\varphi:X\to \mathbb{P}^N$ of irregular surfaces $X$ of general type with at worst canonical singularities, when $\varphi$ is a finite Galois morphism of degree $4$ onto a smooth variety of minimal degree $Y$ inside $\mathbb{P}^N$. These surfaces satisfy $K_X^2 = 4p_g(X)-8$, with $p_g$ an even integer, $p_g \geq 4$. They are classified by the first two authors into four distinct families (three, if $p_g=4$). We show that, when $X$ is general in its family, any deformation of $\varphi$ has degree greater than or equal to $2$ onto its image. More interestingly, we prove that, with two exceptions, a general deformation of $\varphi$ is two--to--one onto its image, which is a surface whose normalization is a ruled surface of appropriate genus. We also show that with the exception of one family, the deformations of a general surface $X$ are unobstructed, and consequently, $X$ belongs to a unique irreducible component of the Gieseker moduli space, which we prove is uniruled (the fourth one being product of curves is well-studied). As a consequence we show the existence of infinitely many moduli spaces with uniruled components corresponding to each even $p_g\geq 4$. Among other things, our results are relevant because they exhibit moduli components such that the degree of the canonical morphism jumps up at proper locally closed subloci. This contrasts with the moduli of surfaces with $K_X^2 = 2p_g - 4$ (which are double covers of surfaces of minimal degree), studied by Horikawa but is similar to the moduli of curves of genus $g\geq 3$.