Let $N=\mathbb{Z}^{2}$, $(a,b)\in N$ with $\gcd(a,b)=1$ and $\iota\colon Q\hookrightarrow N$ the saturated sublattice generated by $(a,b)$. Let $\Sigma$ be the fan in $N_{\mathbb{R}}$ of Example 3.70. Then $X_{\Sigma}=\mathbb{P}^{2}$ with projective coordinates $(x_{0}:x_{1}:x_{2})$. The fan induced in $Q_{\mathbb{R}}$ has three cones: $\Sigma_{Q}=\{\mathbb{R}_{\leq 0},\{0\},\mathbb{R}_{\geq 0}\}$. Thus $X_{\Sigma_{Q}}=\mathbb{P}^{1}$. Let $p=(1:p_{1}:p_{2})$ be a point of $X_{\Sigma,0}(K)$. Then $\varphi_{p,\iota}((1:t))=(1:p_{1}t^{a}:p_{2}t^{b})$. Therefore, $Y_{\Sigma,Q,p}$ is the curve of equation

$$p_{2}^{a}x_{0}^{a}x_{1}^{b}-p_{1}^{b}x_{0}^{b}x_{2}^{a}=0.$$

In general, this curve is not normal. Hence it is not a toric variety.