The Alexander-Hirschowitz theorem and related problems

TL;DR

This paper provides a comprehensive, self-contained proof of the Alexander-Hirschowitz theorem, clarifying when a general set of double points in projective space has the expected Hilbert function, and discusses related open problems.

Contribution

It offers a detailed, accessible proof of the Alexander-Hirschowitz theorem with supplementary appendices and open problems for further research.

Findings

Confirmed the conditions for expected Hilbert functions of double points

Provided a self-contained proof suitable for commutative algebraists

Highlighted open problems in interpolation theory

Abstract

We present a proof of the celebrated result due to Alexander and Hirschowitz which determines when a general set of double points in $\mathbb P^n$ has the expected Hilbert function. Our intended audience are Commutative Algebraists who may be new to interpolation problems. In particular, the main aim of our presentation is to provide a self-contained proof containing all details (including some we could not find in the literature). Also, considering our intended audience, we have added (a) short appendices to make this survey more accessible and (b) a few open problems related to the Alexander-Hirschowitz theorem and the interpolation problems.