Veronese Avoiding Hypersurfaces

TL;DR

This paper introduces Veronese-Avoiding hypersurfaces, characterizes their properties via algebraic and geometric criteria, and explores their parameter space and singularities.

Contribution

It provides a new criterion for Veronese-Avoiding hypersurfaces using Macaulay inverse systems and classifies certain singular hypersurfaces.

Findings

Veronese-Avoiding condition is equivalent to non-degeneracy of the associated form.

Reduced hypersurfaces with n isolated singular points are Veronese-Avoiding iff these are in general linear position.

Identifies a distinguished irreducible nodal locus in the parameter space.

Abstract

We introduce Veronese-Avoiding hypersurfaces, inspired by the theory of associated forms of Alper--Isaev. In the smooth case, we reinterpret their criterion via Macaulay inverse systems: the Veronese-Avoiding condition is equivalent to the non-degeneracy of the associated form. In the singular case, our main theorem shows that a reduced hypersurface with exactly $n$ isolated singular points is Veronese-Avoiding if and only if these points are ordinary nodes in general linear position; we also classify singular plane cubics and treat fewer than $n$ nodes via a natural rational map. We then study the parameter space, proving local closedness and identifying a distinguished irreducible nodal locus. Finally, we prove a Lefschetz-type consequence for the Milnor algebra in degree $1$.