On the Arithmetical Rank of Certain Segre Embeddings

TL;DR

This paper investigates the defining equations of Segre embeddings of projective spaces with hypersurfaces, extends previous results, and explores cohomological dimensions, solving a conjecture in characteristic zero under smoothness assumptions.

Contribution

It extends the understanding of defining equations for Segre products and proves a conjecture relating cohomological and etale cohomological dimensions in characteristic zero.

Findings

Extended results on the arithmetical rank of Segre embeddings.

Proved Lyubeznik's conjecture in characteristic zero for smooth schemes.

Established that a depth-cohomological dimension relationship holds in characteristic zero up to dimension three.

Abstract

We study the number of (set-theoretically) defining equations of Segre products of projective spaces times certain projective hypersurfaces, extending results by Singh and Walther. Meanwhile, we prove some results about the cohomological dimension of certain schemes. In particular, we solve a conjecture of Lyubeznik about an inequality involving the cohomological dimension and the etale cohomological dimension of a scheme, in the characteristic-zero-case and under a smoothness assumption. Furthermore, we show that a relationship between depth and cohomological dimension discovered by Peskine and Szpiro in positive characteristic holds true also in characteristic-zero up to dimension three.