Deformations of finite morphisms and applications to moduli of surfaces of general type with $K^2 = 4p_g-8$

Abstract

In this thesis, 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 $p_g$ being an even number bigger than of equal to $4$. For each $p_g \geq 6$, they are classified in \cite{GP} into four distinct irreducible families (if $p_g=4$, then they are classified into three distinct irreducible families). 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 in addition that, with the exception of one of the families when $p_g=4$ and of another of the families for each $p_g \geq 8$, 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, unless it is a product of genus two curves. In the latter case, it follows that any deformation of $\varphi$ is four--to--one onto its image. We also show that the deformations of a general surface $X$ of three of the four families are unobstructed, and consequently, $X$ belongs to a unique irreducible component of the Gieseker moduli space, which we prove is uniruled. As a consequence of our results we show the existence of the following moduli components containing irregular quadruple Galois canonical covers as proper, locally closed {{subloci}}: \par (1) for any $m \geq 1$ and $p_g = 2m+2$, a $(8m+20)$-dimensional, uniruled, irreducible component of $\mathcal{M}_{8m,1,2m+2}$, whose general element is a canonical double cover of a non-normal surface whose normalization is an elliptic ruled surface with invariant $e = 0$; in particular a general element has a genus $m+1$ fibration over an elliptic curve. \par (2) a $28$-dimensional, uniruled irreducible component of $\mathcal{M}_{16,2,6} $, whose general element is a canonical double cover of a (smooth) ruled surface over a curve of genus $2$ with invariant $e = -2$. \par 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 is in contrast with the moduli of surfaces with $K_X^2 = 2p_g - 4$ (which are double covers of surfaces of minimal degree), studied by Horikawa (see \cite{Hor}) but is pleasingly similar to the moduli of smooth curves of genus $g \geq 3$.