A bound on the degree of schemes defined by quadratic equations

作者：Alzati Alberto^*; Carlos Sierra Jose

来源：Forum Mathematicum, 2012, 24(4): 733-750.

DOI：10.1515/FORM.2011.081

• Varieties defined by quadrics
• 2-Veronese embeddings
• apparent double points
• syzygies

We consider complex projective schemes X subset of P-r 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,N-2 and K-2. In this setting, we show that the degree d of X subset of P-r is bounded by a function of its codimension c, whose asymptotic behavior is given by 2(c) / (4)root pi c, thus improving the obvious bound d %26lt;= 2(c). More precisely, we get the bound ((d)(2)) %26lt;= ((2c-1)(c-1)) . Furthermore, if X satisfies property N-p or N-2,N-p, we obtain the better bound ((d+2-p)(2)) %26lt;= ((2c+3-2p)(c+1-p)). . Some classification results are also given when equality holds.