The projective dimension of three cubics is at most 5

TL;DR

This paper proves the conjecture that for an ideal generated by three degree-three forms in a polynomial ring, the projective dimension is at most 5, confirming computational evidence and improving previous bounds.

Contribution

It establishes the sharp bound of 5 for the projective dimension of ideals generated by three cubics, resolving a conjecture motivated by Stillman's question.

Findings

Proved that the projective dimension is at most 5 for three cubics.

Confirmed the conjecture based on computational evidence.

Improved the known upper bound from 36 to 5.

Abstract

Let $R$ be a polynomial ring over a field and $I$ an ideal generated by three forms of degree three. Motivated by Stillman's question, Engheta proved that the projective dimension $\mathrm{pd}(R/I)$ of $R/I$ is at most 36, although the example with largest projective dimension he constructed has $\mathrm{pd}(R/I)=5$. Based on computational evidence, it had been conjectured that $\mathrm{pd}(R/I)\leq 5$. In the present paper we prove this conjectured sharp bound.