A Tight Bound on the Projective Dimension of Four Quadrics
  Motivated by Stillman's question, we show that the projective dimension of an ideal generated by four quadric forms in a polynomial ring is at most 6; moreover, this bound is tight. We achieve this bound, in part, by giving a characterization of the low degree generators of ideals primary to height three primes of multiplicities one and two. 
Craig Huneke, Paolo Mantero, Jason McCullough, Alexandra Seceleanu
math.AC