3-dimensional sundials

TL;DR

This paper extends the understanding of Hilbert functions for collections of lines in projective space by including 3-dimensional sundials, a special degenerative scheme, thus broadening the class of configurations with known bipolynomial Hilbert functions.

Contribution

The paper generalizes previous results by proving that unions of lines and 3-dimensional sundials also have bipolynomial Hilbert functions in projective spaces of dimension three and higher.

Findings

Unions of lines and sundials have bipolynomial Hilbert functions.

Extension of Hartshorne and Hirschowitz's result to special degenerations.

Provides new insights into the algebraic properties of degenerative schemes.

Abstract

Robin Hartshorne and Alexander Hirschowitz proved that a generic collection of lines on $\mathbb P^n$, $n \geq 3$, has bipolynomial Hilbert Function. We extended this result to a specialization of the collection of generic lines, by considering a union of lines and $3$-dimensional sundials (i.e., a union of schemes obtained by degenerating pairs of skew lines).