Minimal algebraic surfaces of general type with $c^2_1 = 3, p_g = 1 \text{ and } q = 0$, which have non-trivial 3-torsion divisors

Abstract

We shall give a concrete description of minimal algebraic surfaces $X$’s defined over $\mathbb{C}$ of general type with the first chern number 3, the geometric genus 1 and the irregularity 0, which have non-trivial 3-torsion divisors. Namely, we shall show that the fundamental group is isomorphic to $\mathbb{Z}/3$, and that the canonical model of the universal cover is a complete intersection in $\mathbb{P}^{4}$ of type (3, 3).