The projective dimension of codimension two algebras presented by quadrics

Motivated by a question of Stillman, we find a sharp upper bound for the projective dimension of ideals of height two generated by quadrics. In a polynomial ring with arbitrary large number of variables, we prove that ideals generated by n quadrics define cyclic modules with projective dimension at most 2n−2. We refine this bound according to the multiplicity of the ideal. We ask whether tight upper bounds for the projective dimension of ideals generated by quadrics can be expressed only in terms of their height and number of minimal generators.