Connectedness Bertini Theorem via numerical equivalence

TL;DR

This paper provides a new proof of a connectedness theorem for preimages of linear varieties under morphisms from projective varieties, using numerical methods and the Generalized Hodge Index Theorem, applicable in any characteristic.

Contribution

It introduces a novel proof technique for connectedness results leveraging numerical equivalence and the Generalized Hodge Index Theorem, extending applicability across characteristics.

Findings

Preimages of certain linear varieties are connected.

The proof relies on numerical equivalence and the Hodge Index Theorem.

The approach simplifies previous proofs and broadens applicability.

Abstract

Let $X$ be an irreducible projective variety and $f$ a morphism $X \rightarrow \mathbb{P}^n$. We give a new proof of the fact that the preimage of any linear variety of dimension $k\ge n+1-\dim f(X)$ is connected. We prove that the statement is a consequence of the Generalized Hodge Index Theorem using easy numerical arguments that hold in any characteristic. We also prove the connectedness Theorem of Fulton and Hansen as application of our main theorem.