The projective dimension of codimension two algebras presented by quadrics

Abstract

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.

Three themes of syzygies

“…Contrasting with Theorem 4.3, Huneke, Mantero, McCullough, and Seceleanu proved in [HMMS1] that ideals generated by g quadrics with codimension two have projective dimension at most 2g − 2; moreover, known examples show this bound to be optimal. In [HMMS2], they also establish an optimal upper bound of 6 for ideals generated by four quadrics.…”

“…The following example gives the Betti table of one of these ideals: In particular, pd S (S/I) = 6, showing that the bounds in [HMMS1] and [HMMS2] are optimal.…”

Lower bounds on Betti numbers

“…The question of providing effective upper bounds is much harder, and some of the efforts in this direction can be found in [41]. See [42] for an exposition on some of the followup results that expanded on the ideas initiated by Ananyan and Hochster in their proof of Stillman's conjecture; see also [59] for a survey and [50,25] for related work on the subject.…”

On the projective dimension of $5$ quadric almost complete intersections with low multiplicities

“…In this section, we let I be an almost complete intersection generated by 5 quadrics. In [13,Question 6.2] and [13,Question 10.2], the authors asked the following question:…”