# On rigidity of toric varieties arising from bipartite graphs

## TL;DR

This paper characterizes the rigidity of toric varieties derived from bipartite graphs by linking their geometric properties to combinatorial features of the graphs, providing criteria for rigidity based solely on graph structure.

## Contribution

It offers a novel combinatorial characterization of the faces of the cone associated with bipartite graph-derived toric varieties and establishes graph-based criteria for their rigidity.

## Key findings

- Faces of the cone are characterized by independent sets of the bipartite graph.
- Criteria for rigidity are derived purely from graph properties.
- The approach connects combinatorial graph theory with algebraic geometry of toric varieties.

## Abstract

One can associate to a bipartite graph a so-called edge ring whose spectrum is an affine normal toric variety. We characterize the faces of the (edge) cone associated to this toric variety in terms of some independent sets of the bipartite graph. By applying to this characterization the combinatorial study of deformations of toric varieties by Altmann, we present certain criteria for their rigidity purely in terms of graphs.