Surfaces of general type with $p_g=1$, $q=0$, $K^2=6$ and Grassmannians

TL;DR

This paper constructs new algebraic surfaces of general type with specific invariants using Grassmannian-based varieties, linking classical and modern geometric methods.

Contribution

It introduces a novel construction of surfaces with $p_g=1$, $q=0$, $K^2=6$ using Fano fourfolds and Calabi-Yau threefolds related to Grassmannians, connecting to classical Campedelli surfaces.

Findings

Constructed explicit examples of surfaces with $p_g=1$, $q=0$, $K^2=6$.

Linked modern constructions to classical algebraic surfaces via Pfaffian-Grassmannian correspondence.

Expanded the understanding of surfaces of general type with specific invariants.

Abstract

We construct examples of surfaces of general type with $p_g=1$, $q=0$ and $K^2=6$. We use as key varieties Fano fourfolds and Calabi-Yau threefolds that are zero section of some special homogeneous vector bundle on Grassmannians. We link as well our construction to a classical Campedelli surface, using the Pfaffian-Grassmannian correspondence.