Triple planes with p_g=q=0

TL;DR

This paper classifies triple plane surfaces with p_g=q=0 into 12 families, describing their Tschirnhausen bundles, providing explicit constructions, and correcting historical misconceptions about certain types.

Contribution

It introduces a complete classification of these surfaces, details the structure of their Tschirnhausen bundles, and offers new explicit examples including counterexamples to past claims.

Findings

Classified triple planes into 12 families.

Identified bundle structures for each family.

Constructed explicit examples for several types.

Abstract

We show that general triple planes with p_g=q=0 belong to at most 12 families, that we call surfaces of type I,..., XII, and we prove that the corresponding Tschirnhausen bundle is direct sum of two line bundles in cases I, II, III, whereas is a rank 2 Steiner bundle in the remaining cases. We also provide existence results and explicit constructions for surfaces of type I,..., VII, recovering all classical examples and discovering several new ones. In particular, triple planes of type VII provide counterexamples to a wrong claim made in 1942 by Bronowski.