A Family of Ideals with Few Generators in Low Degree and Large Projective Dimension

TL;DR

This paper constructs specific ideals in polynomial rings demonstrating that the projective dimension can grow exponentially with the number and degrees of generators, providing counterexamples to longstanding conjectures.

Contribution

It introduces a family of ideals with few generators in low degree but large projective dimension, answering Zhang's question negatively and offering lower bounds for Stillman's question.

Findings

Projective dimension can grow exponentially with generators' degrees and count.

Counterexamples include an ideal with 7 quadrics and projective dimension 15.

Provides lower bounds for Stillman's conjecture.

Abstract

Stillman posed a question as to whether the projective dimension of a homogeneous ideal I in a polynomial ring over a field can be bounded by some formula depending only on the number and degrees of the minimal generators of I. More recently, motivated by work on local cohomology modules in characteristic p, Zhang asked more specifically if the projective dimension of I is bounded by the sum of the degrees of the generators. We define a family of homogeneous ideals in a polynomial ring over a field of arbitrary characteristic whose projective dimension grows exponentially if the number and degrees of the generators are allowed to grow linearly. We therefore answer Zhang's question in the negative and provide a lower bound to any answer to Stillman's question. We also describe some explicit counterexamples to Zhang's question including an ideal generated by 7 quadrics with projective dimension 15.