A Linear Bound on the Projective Dimension of Height 3 Quadratic Ideals

In 2016, Ananyan and Hochster gave the first proof of a positive answer to Stillman’s Question, which asked for a bound on the projective dimension of a graded polynomial ideal purely in terms of the number and degrees of its generators. Explicit formulas for such a bound are limited and often not optimal. In this paper, we give a nearly optimal linear upper bound on the projective dimension of height $3$ ideals generated by any number of degree $2$ homogenous polynomials.