New surfaces with $K^2=7$ and $p_g=q\leq 2$

TL;DR

This paper constructs new complex surfaces of general type with specific invariants, including cases with $p_g=q=2$, $p_g=q=1$, and $p_g=q=0$, expanding the known examples in algebraic geometry.

Contribution

It introduces explicit constructions of minimal complex surfaces with $K^2=7$ and various $p_g$, $q$ values, including novel double cover methods.

Findings

Constructed surfaces with $p_g=q=2$ and degree 2 Albanese map.

Developed double cover models for surfaces with $p_g=q=1$ and $p_g=q=0$.

Expanded the classification of surfaces with $K^2=7$ and small invariants.

Abstract

We construct smooth minimal complex surfaces of general type with $K^2=7$ and: $p_g=q=2,$ Albanese map of degree $2$ onto a $(1,2)$-polarized abelian surface; $p_g=q=1$ as a double cover of a quartic Kummer surface; $p_g=q=0$ as a double cover of a numerical Campedelli surface with $5$ nodes.