The structure of monotone blow-ups in symplectic toric geometry and a question of McDuff

TL;DR

This paper classifies monotone polytopes in symplectic toric geometry that admit vertex blow-ups, showing only the simplex and its specific blow-up qualify, and characterizes when disjoint face blow-ups are possible, answering a question by McDuff.

Contribution

It provides a complete classification of monotone polytopes allowing certain blow-ups, extending previous results and addressing McDuff's 2011 question.

Findings

Only the simplex and a specific blow-up of it admit vertex blow-ups.

Disjoint face blow-ups occur only when faces are disjoint with dimensions summing to n-1 or n-2.

Results confirm conjectures about the structure of monotone polytopes in symplectic geometry.

Abstract

Monotone polytopes, also known as smooth reflexive polytopes, are the polytopes associated to monotone symplectic toric manifolds and Gorenstein Fano toric varieties. We first show that the only monotone polytopes admitting blow-ups at vertices are the simplex and the result of a codimension-two blow-up in it (this is the polyhedral version of a result of Bonavero from 2002). Then we show that the $n$-simplex admits disjoint blow-ups at faces if and only if the faces are disjoint and have dimensions adding up to $n-1$ or $n-2$. These results answer a question posed by Dusa McDuff in 2011.