Moduli of Canonical Surfaces of General Type with $K_S^2 = 1$, $p_g = 2$

TL;DR

This paper investigates the moduli space of certain algebraic surfaces, establishing its irreducibility, dimension, and unirational compactification, thereby advancing understanding of their classification and geometric properties.

Contribution

It proves the irreducibility, computes the dimension, and constructs a unirational compactification of the moduli space for minimal surfaces with specific invariants.

Findings

Moduli space is irreducible.

Dimension of the moduli space is 28.

Existence of a unirational compactification.

Abstract

We study the moduli space of minimal surfaces of general type with $K_S^2 = 1$ and $p_g = 2$ and show that it is irreducible, has dimension $28$ and admits a compactification which is unirational.