Conic bundles that are not birational to numerical Calabi--Yau pairs

TL;DR

This paper proves that certain general conic bundles over the projective plane with high-degree branch curves are not birational to any variety with an effective multiple of its anticanonical divisor, highlighting their unique birational properties.

Contribution

It establishes non-birationality results for conic bundles with high-degree branch curves and provides examples over number fields, expanding understanding of their birational classification.

Findings

No birational models with effective anticanonical multiples for high-degree branch conic bundles

Examples of 2-dimensional conic bundles over number fields with similar properties

Advances the classification of conic bundles based on their birational and anticanonical properties

Abstract

Let $X$ be a general conic bundle over the projective plane with branch curve of degree at least 19. We prove that there is no normal projective variety $Y$ that is birational to $X$ and such that some multiple of its anticanonical divisor is effective. We also give such examples for 2-dimensional conic bundles defined over a number field.