Postulation of schemes of length at most 4 on surfaces
E. Ballico, S. Canino
TL;DR
Addresses the postulation problem for zero-dimensional schemes of length ≤4 on integral surfaces, classifying length-4 schemes into curvilinear, 2-squares, and tiles, and employing the Horace method with a key 2p3p lemma to relate 2-squares to double points. It proves that general unions have the expected postulation on $\mathbb{P}^2$ and $\mathbb{P}^1\times\mathbb{P}^1$, with extensions to Hirzebruch surfaces via secant-variety non-defectiveness and explicit Hilbert-function formulas for tiles. The results show good postulation holds except for two classical obstructions (two and five double points), and they also identify bad-postulation examples and provide detailed dimension counts for various tile/double-point configurations. Overall, the work clarifies low-length postulation on key surfaces and links local scheme structure to global linear systems and secant-variety geometry.
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.