Unirational quasi-hyperelliptic surfaces in characteristic five

TL;DR

This paper studies unirational quasi-hyperelliptic surfaces in characteristic five, classifying their singular fibers and deriving formulas for their geometric invariants, thereby advancing understanding of these special algebraic surfaces.

Contribution

It provides a classification of singular fibers and formulas for invariants of unirational quasi-hyperelliptic surfaces in characteristic five, including a classification of rational cases.

Findings

Classification of singular fibers in characteristic five

Formulas for arithmetic genus and self-intersection number

Complete classification of rational quasi-hyperelliptic surfaces

Abstract

As a generalization of a quasi-elliptic surface, there is a quasi-hyperelliptic surface, a nonsingular projective surface which has a fibration structure whose general fiber is a quasi-hyperelliptic curve ($=$ singular hyperelliptic curve with one cuspidal singular point) of genus $(p-1)/2$ in characteristic $p \ge 5$. In this paper, we consider unirational quasi-hyperelliptic surfaces in characteristic $5$, and classify singular fibers and give a formula for the arithmetic genus and the self-intersection number of the canonical divisor. As a corollary, we classify rational quasi-hyperelliptic surfaces by determining the combinations of singular fibers, the defining equations and sections.