The Hesse pencil and polarizations of type $(1,3)$ on Abelian surfaces

TL;DR

This paper refines the understanding of discriminant curves on Abelian surfaces with a specific polarization, and demonstrates the existence of a new family of minimal surfaces of general type with particular invariants.

Contribution

It proves a sharper description of the discriminant curve's singularities and establishes the existence of a new family of minimal surfaces with specified properties.

Findings

Discriminant curve of degree 18 with 36 nodes and 72 cusps.

Degeneration behavior of the discriminant curve as the surface approaches a product of elliptic curves.

Existence of a new family of minimal surfaces with p_g=q=2, K^2=6, and degree 3 Albanese map.

Abstract

In this short note we prove two theorems, the first one is a sharpening of a result of Lange and Sernesi: the discriminant curve W of a general Abelian surface $A$ endowed with an irreducible polarization $D$ of type $(1,3)$ is an irreducible curve of degree $18$ whose singularities are exactly $36$ nodes and $72$ cusps. Moreover, we analyze the degeneration of the discriminant curve $W$ and its singularities as $A$ tends to the product of two equal elliptic curves. The second theorem, using the first one in order to prove a transversality assertion, shows that the general element of a family of surfaces constructed by Alessandro and Catanese is a smooth surface, thereby proving the existence of a new family of minimal surfaces of general type with $p_g=q=2, K^2=6$ and Albanese map of degree $3$.