Bound on the projective dimension of three cubics

TL;DR

This paper establishes an upper bound of 36 on the projective dimension for ideals generated by three cubic forms in a polynomial ring, and constructs an example with projective dimension 5, answering a longstanding question.

Contribution

It provides the first explicit bound on the projective dimension for three cubic generators and constructs a counterexample with dimension 5.

Findings

Projective dimension of three cubics is at most 36.

Existence of an ideal generated by three cubics with projective dimension 5.

Answer to whether projective dimension can exceed 4 is yes.

Abstract

We show that given any polynomial ring R over a field, and any ideal J in R which is generated by three cubic forms, the projective dimension of R/J is at most 36. We also settle the question whether ideals generated by three cubic forms can have projective dimension greater than 4, by constructing one with projective dimension equal to 5.