Matrixwise (approach to unexpected hypersurfaces) Reloaded

TL;DR

This paper introduces a novel combinatorial approach to prove the existence of specific hypersurfaces with prescribed multiplicities and vanishing conditions, advancing the understanding of unexpected hypersurfaces often studied through computational experiments.

Contribution

It presents a new method leveraging weak combinatorics to establish hypersurface existence, providing a theoretical tool for the study of unexpected hypersurfaces.

Findings

New combinatorial proof technique for hypersurface existence

Application to unexpected hypersurfaces based on computational examples

Enhanced theoretical understanding of hypersurface conditions

Abstract

In the paper we provide a new method of proving the existence of a hypersurface of degree $d$ in $\mathbb{P}^n$, with a general point of multiplicity $m$ and vanishing at a given set of points $Z$, by looking at weak combinatorics of a set $Z$. This method has a direct application in the theory of unexpected hypersurfaces, where many of the examples are based only on computer experiments.