# Singularities and radical initial ideals

## TL;DR

This paper investigates which reduced monomial schemes can arise as Gr"obner degenerations of smooth projective varieties, proposing a conjecture linking them to Stanley-Reisner schemes of acyclic Cohen-Macaulay complexes and proving it in specific cases.

## Contribution

It formulates a conjecture characterizing possible degenerations and proves it for certain orders, curves, polytopes, and graphs, advancing understanding of degenerations and singularities.

## Key findings

- Conjecture holds for degrevlex orders.
- Conjecture verified for elliptic curves over real fields.
- Conjecture confirmed for boundaries of cross-polytopes and leafless graphs.

## Abstract

What kind of reduced monomial schemes can be obtained as a Gr\"obner degeneration of a smooth projective variety? Our conjectured answer is: only Stanley-Reisner schemes associated to acyclic Cohen-Macaulay simplicial complexes. This would imply, in particular, that only curves of genus zero have such a degeneration. We prove this conjecture for degrevlex orders, for elliptic curves over real number fields, for boundaries of cross-polytopes, and for leafless graphs. We discuss consequences for rational and F-rational singularities of algebras with straightening laws.