A family of ideals with few generators in low degree and large projective dimension.

Stillman posed a question as to whether the projective dimension of a homogeneous ideal $ I$. More recently, motivated by work on local cohomology modules in characteristic $ p$ 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. [ABSTRACT FROM AUTHOR]