Postulation of schemes of length at most 4 on surfaces

TL;DR

This paper investigates the postulation problem for zero-dimensional schemes of length up to 4 on various surfaces, providing general results and specific cases for P2, P1xP1, and Hirzebruch surfaces.

Contribution

It establishes new results on the postulation of small schemes on surfaces, including general theorems and specific cases for key surfaces.

Findings

Most small schemes have good postulation in P2 and P1xP1

Exceptions are limited to well-known special cases

Results extend to Hirzebruch surfaces

Abstract

In this paper we address the postulation problem of zero-dimensional schemes on a surface of length at most 4. We prove some general results and then we focus on the case of P2, P1xP1 and Hirzebruch surfarces. In particular, we prove that except for few well-known exceptions, a general union of schemes of length at most 4 has always good postulation in P2 and in P1xP1.