The Minimal Resolution Conjecture on a general quartic surface in $\mathbb P^3$

TL;DR

This paper proves the Minimal Resolution Conjecture for a general quartic surface in projective 3-space, extending previous results for quadric and cubic surfaces using Gorenstein liaison and dimension computations.

Contribution

It establishes the conjecture for quartic surfaces in P^3, employing Gorenstein liaison and dimension counting techniques to demonstrate the existence of certain algebraic links.

Findings

Proves the conjecture for general quartic surfaces in P^3.

Uses Gorenstein liaison and dimension computations as key tools.

Discusses limitations in higher degree cases.

Abstract

Musta\c{t}\u{a} has given a conjecture for the graded Betti numbers in the minimal free resolution of the ideal of a general set of points on an irreducible projective algebraic variety. For surfaces in $\mathbb P^3$ this conjecture has been proven for points on quadric surfaces and on general cubic surfaces. In the latter case, Gorenstein liaison was the main tool. Here we prove the conjecture for general quartic surfaces. Gorenstein liaison continues to be a central tool, but to prove the existence of our links we make use of certain dimension computations. We also discuss the higher degree case, but now the dimension count does not force the existence of our links.