On the Hilbert function of a finite scheme contained in a quadric surface (Q6617344)

This paper concerns the number of linearly independent curves passing through a finite scheme of length \(l\), of degree at least \(l- 2\), contained in a smooth quadric surface over the complex numbers.\N\NThe author, after recalling the known results on the Hilbert function of zero-dimensional multiprojective schemes and reviewing the geometry of the punctual Hilbert scheme of a surface (see Section 2 and Section 3), proves vanishing theorems for \(H^1(\mathcal I_Z(m,n))\) (see Theorem 4.4 and Theorem 5.2), these will be used, in Section 6, to describe the Brill-Noether loci in Hilb(\(l\)); cf. Proposition 6.6. In Section 7, the Euler characteristic of flag Hilbert schemes is computed.