Special triple covers of algebraic surfaces

TL;DR

This paper investigates special triple covers of algebraic surfaces with particular Tschirnhausen bundles, providing criteria for their existence and classifying special triple planes, including families of K3 surfaces.

Contribution

It introduces new criteria for the existence of such covers and classifies special triple planes, revealing new families of K3 surfaces.

Findings

Criteria for existence of special triple covers

Complete classification of special triple planes

Identification of new K3 surface families

Abstract

We study special triple covers $f\colon T \to S$ of algebraic surfaces, where the Tschirnhausen bundle $\mathcal E = \left(f_*\mathcal O_T/\mathcal O_S\right)^\vee$ is a quotient of a split rank three vector bundle, and we provide several necessary and sufficient criteria for the existence. As an application, we give a complete classification of special triple planes, finding among others two nice families of K3 surfaces.