Toric Degenerations of Low Degree Hypersurfaces

TL;DR

This paper characterizes when low degree hypersurfaces in projective space can be degenerated into toric varieties using Gr"obner bases, establishing a precise degree bound.

Contribution

It provides a complete criterion for toric Gr"obner degenerations of hypersurfaces based on their degree relative to the dimension.

Findings

Hypersurfaces of degree d ≤ 2n-1 admit toric Gr"obner degenerations.

The degree bound is both necessary and sufficient.

The result clarifies the relationship between hypersurface degree and degenerability into toric varieties.

Abstract

We show that a sufficiently general hypersurface of degree $d$ in $n$-dimensional projective space admits a toric Gr\"obner degeneration if and only if $d\leq 2n-1$.