On the arithmetical rank of a special class of minimal varieties

TL;DR

This paper investigates the arithmetical ranks and cohomological dimensions of a broad class of Cohen-Macaulay varieties of minimal degree, revealing instances of set-theoretic complete intersections and cases with high arithmetical rank.

Contribution

It provides new insights into the arithmetical ranks of minimal degree varieties, including examples with arbitrarily large arithmetical rank compared to codimension.

Findings

Existence of infinitely many set-theoretic complete intersections.

Examples where arithmetical rank exceeds codimension arbitrarily.

Analysis of cohomological dimensions of these varieties.

Abstract

We study the arithmetical ranks and the cohomological dimensions of an infinite class of Cohen-Macaulay varieties of minimal degree. Among these we find, on the one hand, infinitely many set-theoretic complete intersections, on the other hand examples where the arithmetical rank is arbitrarily greater than the codimension.