Combinatorial bounds on Hilbert functions of fat points in projective space

TL;DR

This paper establishes combinatorial bounds for Hilbert functions of fat point schemes in projective space, providing explicit formulas and criteria for equality, with applications to graded Betti numbers and various examples.

Contribution

It introduces combinatorial bounds for Hilbert functions of fat points, including explicit formulas in the plane and criteria for their exactness, extending previous algebraic results.

Findings

Explicit bounds for N=2 case

Criterion for bounds to be equal

Exact Hilbert functions and Betti numbers for many examples

Abstract

We study Hilbert functions of certain non-reduced schemes A supported at finite sets of points in projective space, in particular, fat point schemes. We give combinatorially defined upper and lower bounds for the Hilbert function of A using nothing more than the multiplicities of the points and information about which subsets of the points are linearly dependent. When N=2, we give these bounds explicitly and we give a sufficient criterion for the upper and lower bounds to be equal. When this criterion is satisfied, we give both a simple formula for the Hilbert function and combinatorially defined upper and lower bounds on the graded Betti numbers for the ideal defining A, generalizing results of Geramita-Migliore-Sabourin (2006). We obtain the exact Hilbert functions and graded Betti numbers for many families of examples, interesting combinatorially, geometrically, and algebraically. Our method works in any characteristic. AWK scripts implementing our results can be obtained at http://www.math.unl.edu/~bharbourne1/CHT/Example.html .