On the Hilbert Function of a Finite Scheme Contained in a Quadric Surface

TL;DR

This paper investigates the Hilbert function of finite schemes on smooth quadric surfaces, specifically determining the number of independent curves of certain degrees passing through the scheme.

Contribution

It provides a precise calculation of the Hilbert function for finite schemes on smooth quadric surfaces, extending understanding of algebraic geometry in this context.

Findings

Calculated the number of independent curves of degree ≥ l - 1 passing through the scheme.

Established a formula for the Hilbert function of finite schemes in smooth quadric surfaces.

Enhanced the understanding of the algebraic properties of schemes on quadrics.

Abstract

Consider a finite scheme of length l contained in a smooth quadric surface over the complex numbers. We determine the number of linearly independent curves passing through the scheme, of degree at least l - 1.