An Unbounded Family of log Calabi-Yau Pairs

TL;DR

This paper constructs an explicit family of log Calabi-Yau pairs with specific properties, using a sequence of blow-ups and coverings over projective bundles on Segre-Hirzebruch surfaces, highlighting new geometric examples.

Contribution

It introduces a novel explicit construction of log Calabi-Yau pairs with decreasing Euler characteristic, expanding the known examples in algebraic geometry.

Findings

Constructed explicit examples of log Calabi-Yau pairs with decreasing Euler characteristic.

Demonstrated the existence of such pairs over a sequence of blow-ups.

Provided a detailed geometric construction involving coverings and blow-ups.

Abstract

We give an explicit example of log Calabi-Yau pairs that are log canonical and have a linearly decreasing Euler characteristic. This is constructed in terms of a degree two covering of a sequence of blow ups of three dimensional projective bundles over the Segre-Hirzebruch surfaces ${\mathbb F}_n$ for every positive integer $n$ big enough.