Interpolation in Weighted Projective Spaces

TL;DR

This paper extends the classical double point interpolation problem to weighted projective spaces, establishing foundational algebraic results and demonstrating cases where the Alexander-Hirschowitz theorem analogously applies.

Contribution

It develops the algebraic groundwork for interpolation in weighted projective spaces and proves the theorem's analogue in specific cases.

Findings

Hilbert function of general points behaves as expected in weighted projective spaces

An inductive procedure similar to Terracini's method is established for weighted spaces

An interpolation bound for an infinite family of weighted projective planes is provided

Abstract

Over an algebraically closed field, the $\textit{double point interpolation}$ problem asks for the vector space dimension of the projective hypersurfaces of degree $d$ singular at a given set of points. After being open for 90 years, a series of papers by J. Alexander and A. Hirschowitz in 1992--1995 settled this question in what is referred to as the Alexander-Hirschowitz theorem. In this paper we primarily use commutative algebra to lay the groundwork necessary to prove analogous statements in the $\textit{weighted projective space}$, a natural generalization of the projective space. We show the Hilbert function of general simple points in any $n$-dimensional weighted projective space exhibits the expected behavior. We give an inductive procedure for weighted projective space, similar to that originally due to A. Terracini from 1915, to demonstrate an example of a weighted projective plane where the analogue of the Alexander-Hirschowitz theorem holds without exceptions. We further adapt Terracini's lemma regarding secant varieties to give an interpolation bound for an infinite family of weighted projective planes.