A two-dimensional family of surfaces of general type with $p_g=0$ and $K^2=7$

TL;DR

This paper constructs a new two-dimensional family of minimal smooth surfaces of general type with invariants p_g=0 and K^2=7, expanding known examples by using Galois covers over rational surfaces with specific properties.

Contribution

It introduces a novel two-dimensional family of surfaces with p_g=0 and K^2=7, constructed as Galois covers over rational surfaces with eight nodes and elliptic fibrations.

Findings

Constructed a two-dimensional family of surfaces with p_g=0 and K^2=7.

Surfaces are Galois $bZ_2 imes bZ_2$-covers over rational surfaces.

Family differs from previously known examples.

Abstract

We study the construction of complex minimal smooth surfaces $S$ of general type with $p_g(S)=0$ and $K_S^2=7$. Inoue constructed the first examples of such surfaces, which can be described as Galois $\mathbb{Z}_2\times\mathbb{Z}_2$-covers over the four-nodal cubic surface. Later the first named author constructed more examples as Galois $\mathbb{Z}_2\times\mathbb{Z}_2$-covers over certain six-nodal del Pezzo surfaces of degree one. In this paper we construct a two-dimensional family of minimal smooth surfaces of general type with $p_g=0$ and $K^2=7$, as Galois $\mathbb{Z}_2\times\mathbb{Z}_2$-covers of certain rational surfaces with Picard number three, with eight nodes and with two elliptic fibrations. This family is different from the previous ones.