Non-elementary Fano conic bundles

TL;DR

This paper investigates non-elementary Fano conic bundle contractions, establishing bounds on the difference in Picard numbers and providing new geometric insights, especially under specific factoriality and singularity conditions.

Contribution

It proves upper bounds on the Picard number difference for non-elementary Fano conic bundles and explores geometric properties under various singularity assumptions.

Findings

The Picard number difference is at most 8 for non-elementary Fano conic bundles.

For certain factorial and singularity conditions, the difference is at most 9.

New geometric information about the structure of these varieties is derived.

Abstract

We study a particular kind of fiber type contractions between complex, projective, smooth varieties f:X->Y, called Fano conic bundles. This means that X is a Fano variety, and every fiber of f is isomorphic to a plane conic. Denoting by rho_{X} the Picard number of X, we investigate such contractions when rho_{X}-rho_{Y} is greater than 1, called non-elementary. We prove that rho_{X}-rho_{Y} is at most 8, and we deduce new geometric information about our varieties, depending on rho_{X}-rho_{Y}. Moreover, when X is locally factorial with canonical singularities and with at most finitely many non-terminal points, we consider fiber type K_{X}-negative contractions f:X->Y with one-dimensional fibers, and we show that rho_{X}-rho_{Y} is at most 9.