New examples of Calabi-Yau threefolds and genus zero surfaces

TL;DR

This paper classifies certain automorphism subgroups of a product of projective lines and uses this to construct new examples of Calabi-Yau threefolds with small Hodge numbers and a novel family of minimal surfaces of general type.

Contribution

It provides a classification of automorphism subgroups leading to new Calabi-Yau threefolds and minimal surfaces, expanding known examples and understanding of their moduli.

Findings

New Calabi-Yau threefolds with Picard number 1 and 5 moduli.

A minimal surface of general type with K^2=3 and fundamental group of order 16.

The constructed family dominates a 4-dimensional component of the moduli space.

Abstract

We classify the subgroups of the automorphism group of the product of 4 projective lines admitting an invariant anticanonical smooth divisor on which the action is free. As a first application, we describe new examples of Calabi-Yau 3-folds with small Hodge numbers. In particular, the Picard number is 1 and the number of moduli is 5. Furthermore, the fundamental group is non-trivial. We also construct a new family of minimal surfaces of general type with geometric genus zero, K^2=3 and fundamental group of order 16. We show that this family dominates an irreducible component of dimension 4 of the moduli space of the surfaces of general type.