In this nicely written article, the author studies rigid toric varieties that arise from bipartite graphs. Let \(G\) be a simple graph with \(V(G)\) the set of its vertices and \(E(G)\) the set of edges. We define the edge ring to \(G\) as \[\mathrm{Edr}(G):= \mathbb{C}[t_{i}t_{j} \, : \, \{i,j\} \in E(G)\,, i,j \in V(G)\}.\] Consider the surjective ring morphism \[\mathbb{C} [x_{e} \, : \, e \in E(G)] \rightarrow\mathrm{Edr}(G)\] \[x_{e} \mapsto t_{i}t_{j},\] where \(e = \{i,j\} \in E(G)\). The kernel \(I_{G}\) of this morphism is the well-known edge ideal. The associated toric variety to the graph \(G\) is defined as \[\mathrm{TV}(G) : =\mathrm{Spec}(\mathbb{C}[x_{e} \, : \, e \in E(G) ] \, \backslash \, I_{G}) =\mathrm{Spec}(\mathbb{C}[\sigma^{\vee}_{G}\cap M]),\] where \(\sigma^{\vee}_{G}\) is called the dual edge cone. The edge ring \(\mathrm{Edr}(G)\) is an integrally closed domain and hence \(\mathrm{ TV}(G)\) is a normal variety. In the present paper the author focuses on the case where \(G\) is a bipartite graph and the main goal of the paper is to understand the first order deformations of \(\mathrm{TV}(G)\). The main results of the paper provide certain criteria for the bipartite graph \(G\) such that the first oder deformation of \(\mathrm{TV}(G)\) are all trivial, which means that \(\mathrm{TV}(G)\) is rigid. In the first step, the author describes the edge cone \(\sigma_{G}\) associated to the toric variety \(\mathrm{TV}(G)\). More precisely, the author shows a one-to-one correspondence between the set of extremal ray generators of the cone \(\sigma_{G}\) and the so-called first independent set (see Definition 2.7 therein). Using this approach one can determine the faces of \(\sigma_{G}\). For this, one defines a spanning subgraph \(G\{A\} \subset G\) associated to the first independent set \(A\). It turns out that a set \(S\) of \(d\) first independent sets (or equivalently, a set of \(d\) extremal rays of \(\sigma_{G}\)) spans a face of dimension \(d\) iff \(\bigcap_{A\in S}G\{A\}\) has \(d+1\) connected components. This result allows to prove that \(\mathrm{TV}(G)\) is smooth in codimension \(2\)
The first main result of the paper, devoted to the rigidity, can be formulated as follows.
Theorem A. Let \(G \subseteq K_{n,m}\) be a connected bipartite graph. Assume that the edge cone \(\sigma_{G}\) admits a three-dimensional non-simplicial face. Then \(\mathrm{TV}(G)\) is not rigid.
Moreover, we present the characterization of rigid affine toric varieties \(\mathrm{TV}(G)\), where \(G\) has exactly one two-side first independent set \(C = C_{1} \sqcup C_{2}\). In other words, one considers the connected bipartite graphs \(G \subseteq K_{n,m}\), where we remove all the edges between two vertex sets \(\emptyset \neq C_{1}\subseteq U_{1}\) and \(\emptyset \neq C_{2} \subseteq U_{2}\), where \(U_{1},U_{2}\) are disjoint sets of the bipartite graph \(G\).
Theorem B. Let \(G \subseteq K_{n,m}\) be a connected bipartite graph with exactly one two-side first independent set \(C \in \mathcal{I}^{1}_{G}\). Then
i) \(\mathrm{TV}(G)\) is not rigid if \(|C_{1}|=1\) and \(|C_{2}| = n-2\), or if \(|C_{1}| = m-2\) and \(|C_{2}| = 1\);
ii) \(\mathrm{TV}(G)\) is rigid, otherwise.