The moduli space of even surfaces of general type with K^2 = 8, p_g = 4 and q = 0

TL;DR

This paper classifies the moduli space of even surfaces of general type with specific invariants, revealing its structure as two intersecting irreducible components and providing a detailed description of their canonical rings.

Contribution

It provides the first classification of these surfaces using moduli theory, identifying two main components of the moduli space and analyzing their properties.

Findings

The moduli space has two 35-dimensional irreducible components intersecting in a codimension one subset.

Surfaces in the second component have always singular canonical models.

Complete description of the half-canonical rings of the surfaces.

Abstract

We settle the first step for the classification of surfaces of general type with K^2 = 8, p_g = 4 and q = 0, classifying the even surfaces (K is 2-divisible). The first even surfaces of general type with $K^2=8$, $p_g=4$ and $q=0$ were found by Oliverio as complete intersections of bidegree (6,6) in a weighted projective space P(1,1,2,3,3). In this article we prove that the moduli space of even surfaces of general type with K^2 = 8, p_g = 4 and q = 0 consists of two 35 -dimensional irreducible components intersecting in a codimension one subset (the first of these components is the closure of the open set considered by Oliverio). For the surfaces in the second component the canonical models are always singular, hence we get a new example of generically nonreduced moduli spaces. Our result gives a posteriori a complete description of the half-canonical rings of the above even surfaces. The method of proof is, we believe, the most interesting part of the paper. After describing the graded ring of a cone we are able, combining the explicit description of some subsets of the moduli space, some deformation theoretic arguments, and finally some local algebra arguments, to describe the whole moduli space. This is the first time that the classification of a class of surfaces can only be done using moduli theory: up to now first the surfaces were classified, on the basis of some numerical inequalities, or other arguments, and later on the moduli spaces were investigated.