  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.
