I believe I may have a counterexample to Theorem 5.3.1 on page 179 from the book book *Discriminants, Resultants and Multidimensional Determinants* by Gel'fand, Kapranov, and Zelevinsky. To summarize the notation (see the section "Notation from the book" below for more details):

- $A$ is finite;
- $X_A$ is the (not necessarily normal) projective toric variety induced by $A$;
- $S$ is the affine semigroup generated $A\times \{1\}$ in $\Xi=\Bbb Z^k$;
- $S/\Gamma$ is the image of $S$ in $\Xi/(\Xi \cap \operatorname{Lin}_\Bbb R(\Gamma))$, where $\operatorname{Lin}_{\Bbb R}(\Gamma)$ denotes the $\Bbb R$-linear span of $\Gamma\times\{1\}$; and
- $i(\Gamma, A) = [\Xi\cap \operatorname{Lin}_\Bbb R(\Gamma) : \operatorname{Lin}_\Bbb Z (A\cap \Gamma)]$.

Here's the theorem in question: The only part of the theorem that I have issue with is the final sentence, and I believe a counterexample to this part.

# Counterexample?

Let \begin{equation*} A = \begin{bmatrix} 1 &0 &0 &1 &0\\ 0 &1 &0 &1 &0\\ 0 &0 &2 &1 &0\\ 1 &1 &1 &1 &1 \end{bmatrix} \subseteq \Xi, \qquad \Gamma = \operatorname{conv}(\omega^{(3)},\omega^{(5)}). \end{equation*} (I'm identifying $A$ with the set of its columns). Note that each column of $A$ is a vertex of $Q=\operatorname{conv}(A)$, and $Q$ is a simplicial polytope. Hence, $\Gamma$ is indeed a face of $Q$.

Claim:The local structure of $X_A$ along the orbit $X^0(\Gamma)$ corresponding to $\Gamma$ is two copies of $\operatorname{Spec}(S/\Gamma) \times X^0(\Gamma)$ glued along a subvariety of dimension $2=1 + \dim(X^0(\Gamma))$. This contradicts the claim in the Theorem that the two copies of $\operatorname{Spec}(S/\Gamma)$ are glued along $\{0\}\times X^0(\Gamma)$.

**Proof.** I'm going to start along a similar route as in the proof given in the book. Let $Y_A=\operatorname{Spec}\Bbb C[S]$---this is the cone over $X_A$. Let $Y^0(\Gamma)\subseteq Y_A$ be the cone over the orbit of $X^0(\Gamma)$. As the book states, "[t]he local structure of $Y_A$ along $Y^0(\Gamma)$ is the same as that of $X_A$ along $X^0(\Gamma)$ so we shall study $Y_A$."

*Outline of proof:* We first show that the local structure of the orbit $Y^0(\Gamma)$ near the distinguished point is two copies of $\Bbb C^2\times Y^0(\Gamma)$ glued along a subvariety of dimension $3=1 + \dim(Y^0(\Gamma))$. We then show that $\operatorname{Spec}(S/\Gamma)\cong \Bbb C^2$.

*The proof itself:* The toric ideal of $A$ is $I_A= (x^2y^2z - u^2v^3)\subseteq \Bbb C[x,y,z,u,v]$. Let $P$ be the distinguished point of the orbit $Y^0(\Gamma)$ in $Y_A$ corresponding to $\Gamma$ ($P$ is called $y_0$ in the book). Let
$$ U = \operatorname{Spec}\Bbb C[\operatorname{Lin}_\Bbb Z(A\cap \Gamma) + S] = \operatorname{Spec}(\Bbb C[x,y,z^\pm,u,v^\pm]/(x^2y^2z - u^2v^3))$$
be the smallest torus-invariant open subset of $Y_A$ containing $Y^0(\Gamma)$ (in the book, $U$ is called $Z$). Consider the projection
$$p\colon U \to Y^0(\Gamma)$$
induced by the inclusion $\operatorname{Lin}_\Bbb Z(A\cap \Gamma)\hookrightarrow \operatorname{Lin}_\Bbb Z(A\cap \Gamma) + S$. We have
\begin{align*}
p^{-1}(P) &= \operatorname{Spec} \bigl(\Bbb C[\operatorname{Lin}_\Bbb Z(A\cap \Gamma) + S]/(z-1, v-1) \bigr)\\
&= \operatorname{Spec} \bigl(\Bbb C[x,y,z^\pm,u,v^\pm]/(z-1, v-1, x^2y^2z - u^2v^3)\bigr)\\
&\cong \operatorname{Spec} \bigl(\Bbb C[x,y,u]/(x^2y^2-u^2)\bigr)\\
&= V(xy-u)\cup V(xy+u) \subseteq \Bbb C^3.
\end{align*}
(Note that $V(xy-u)\cong V(xy+u)\cong \Bbb C^2$). In addition,
$$V(xy-u)\cap V(xy+u) = V(xy-u, xy+u) = V(xy,u) = V(x,u)\cup V(y, u)$$
is the union of two lines---in particular, it has dimension greater than 0. So, $p^{-1}(P)$ is the union of two irreducible varieties glued along a variety of dimension greater than 0.

We now show that these two varieties are isomorphic to $\operatorname{Spec} \Bbb C[S/ \Gamma]$. Because $V(xy-u)\cong V(xy+u)$, it suffices to show that $\Bbb C[S/\Gamma] \cong \Bbb C[x,y,u]/(xy-u)$.

By Macaulay2, we have (with $t$ corresponding to the point $(0,0,1,1)$) \begin{align*} \Bbb C[S + \Xi\cap \Bbb R_{\geq0}(\Gamma\times \{0\})] &= \Bbb C[x,y,z,u,v,t]/(t^2-zv, tuv-xyz, txy-uv^2). \end{align*} Then \begin{align*} \Bbb C[S/\Gamma] &= \Bbb C[S + \Xi \cap \operatorname{Lin}_\Bbb R(\Gamma)]/(z-1, v-1, t-1)\\ &= \Bbb C[x,y,z^\pm, u, v^\pm, t^\pm]/ (t^2-zv, tuv-xyz, txy-uv^2, z-1, v-1, t-1)\\ &= \Bbb C[x,y,u]/(u-xy). \qquad\qquad \Box \end{align*}

# Notation from the book

Here are pictures of the relevant notation from the book. The only things missing is the definition of $X_A$ and $X^0(\Gamma)$. If $A=\{\omega^{(1)},\ldots,\omega^{(n)}\}$, then $X_A$ is the closure in $\Bbb P^{n-1}$ of the set $$ X_A^0 = \{(x^{\omega^{(1)}},\ldots,x^{\omega^{(n)}}) : x = (x_1,\ldots,x_{k-1})\in (\Bbb C^*)^{k-1}\},$$ and $$ X^0(\Gamma) = \{(z_1,\ldots,z_n) \in X_A : z_i=0 \text{ iff }\omega^{(i)}\notin \Gamma\}.$$