Degenerations to Unobstructed Fano Stanley-Reisner Schemes

TL;DR

This paper constructs degenerations of certain algebraic varieties to unobstructed Fano Stanley-Reisner schemes, enabling new toric degenerations and revealing properties of boundary complexes related to the associahedron.

Contribution

It introduces a method to degenerate Mukai varieties to unobstructed Fano Stanley-Reisner schemes and explores their higher-dimensional analogs, including Calabi-Yau cases.

Findings

Degenerations to unobstructed Fano Stanley-Reisner schemes are constructed.

The Stanley-Reisner ring of the associahedron's dual polytope boundary complex has trivial T^2.

New toric degenerations of Fano threefolds are identified.

Abstract

We construct degenerations of Mukai varieties and linear sections thereof to special unobstructed Fano Stanley-Reisner schemes corresponding to convex deltahedra. This can be used to find toric degenerations of rank one index one Fano threefolds. In the second part we find many higher dimensional unobstructed Fano and Calabi-Yau Stanley-Reisner schemes. The main result is that the Stanley-Reisner ring of the boundary complex of the dual polytope of the associahedron has trivial $T^2$.