Postulation of general quartuple fat point schemes in P^3

TL;DR

This paper investigates the algebraic properties of general quartuple fat point schemes in projective 3-space, establishing their expected postulation in high degrees and exploring low-degree cases with computational tools.

Contribution

It proves that such schemes have the expected postulation for degrees 41 and above, extending understanding of fat point schemes in algebraic geometry.

Findings

Expected postulation confirmed for degrees ≥41

Low-degree cases analyzed with computer algebra

Provides new insights into fat point schemes in P^3

Abstract

We study the postulation of a general union $Y$ of double, triple, and quartuple points of $\mathbb{P}^3$. We prove that $Y$ has the expected postulation in degree $d\ge 41$, using the Horace differential lemma. We also discuss the cases of low degree with the aid of computer algebra.