What question did this study set out to answer?

The aim is to calculate the Hilbert function and the graded Betti numbers of the second symbolic power of a configuration in projective space.

February 12, 2026

The Hilbert function and the graded Betti numbers of the 2-nd symbolic power of a 𝕜-configuration in ℙ2

Key Points

The aim is to calculate the Hilbert function and the graded Betti numbers of the second symbolic power of a configuration in projective space.
Derived calculations for Hilbert functions and graded Betti numbers.
Analyzed cases based on the linearity of the reduction vector.
Defined a complete intersection configuration for improved analysis.
Identified conditions under which the graded Betti numbers are calculated for specific configurations.
Provided a complete answer for graded Betti numbers when certain conditions about lines and points are met.
Obtained the Hilbert function for cases where the reduction vector is not GMS.

Abstract

In 2011, Cooper, Harbourne, and Teitler pointed out that the Hilbert function of a Formula: see text-dimensional subscheme (not necessarily reduced) in Formula: see text having a GMS reduction vector is uniquely determined, but the graded Betti numbers are not uniquely determined. In this paper, we find how to calculate the Hilbert function or a graded minimal free resolution of the Formula: see text-nd symbolic power Formula: see text of a Formula: see text-configuration Formula: see text in Formula: see text of type Formula: see text with Formula: see text. For Formula: see text, we find a complete answer for the graded Betti numbers for Formula: see text when the number of lines containing the maximum possible number of points is greater than Formula: see text, and the Hilbert function when Formula: see text is linear, namely, only one line contains Formula: see text points, and the reduction vector for Formula: see text is NOT GMS. For Formula: see text, we find the graded Betti numbers for Formula: see text when Formula: see text is linear, i.e., the reduction vector for Formula: see text is NOT GMS. For Formula: see text, we find the Hilbert function for Formula: see text when Formula: see text is linear, i.e., the reduction vector for Formula: see text is NOT GMS. We also define a complete intersection Formula: see text-configuration in Formula: see text, which is a useful tool to find the Hilbert function or a graded minimal free resolution of the Formula: see text-nd symbolic power of a Formula: see text-configuration in Formula: see text. We also find a graded minimal free resolution of a non-linear Formula: see text-configuration in Formula: see text of type Formula: see text with Formula: see text, whose reverse reduction vector Formula: see text is NOT GMS.

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