Connectivity of hyperplane sections of domains

TL;DR

This paper investigates a problem in commutative algebra related to the connectivity properties of hyperplane sections of domains, challenging previous assumptions and showing that certain principles do not hold, leaving the main problem open.

Contribution

It demonstrates that the principle suggesting the non-existence of such a ring is false, thus providing new insights into the structure of hyperplane sections of domains.

Findings

The principle leading to the non-existence of the ring is false.

The specific problem remains open.

Provides a new perspective on the connectivity of hyperplane sections.

Abstract

During the conference held in 2017 in Minneapolis for his 60th birthday, Gennady Lyubeznik proposed the following problem: Find a complete local domain and an element in it having three minimal primes such that the sum of any two of them has height 2 and the sum of the three of them has height 4. In this note this beautiful problem will be discussed, and will be shown that the principle leading to the fact that such a ring cannot exist is false. The specific problem, though, remains open