A bound on the degree of schemes defined by quadratic equations

We consider complex projective schemes X C Pr defined by quadratic equations and satisfying a technical hypothesis on the fibres of the rational map associated to the linear system of quadrics defining X. Our assumption is related to the syzygies of the defining equations and, in particular, it is weaker than properties N 2, N 2;2 and K 2. In this setting, we show that the degree d of X C Pr is bounded by a function of its codimension c, whose asymptotic behavior is given by 2c=p4 2c, thus improving the obvious bound d ≤ 2 c. More precisely, we get the bound d 2 c ≤ (2c-1 c-1 ) Furthermore, if X satisfies property Np or N2;p, we obtain the better bound (d+2-p 2) ≤ (2c+3-2p c+1-p) Some classification results are also given when equality holds.