Pome

Example 21.13.12C51FE704A3B3

Example 21.13.12.C51FE704A3B3 In this case, $\Sigma $ is the fan in 21.13.13 consisting of three $2$-dimensional cones $\sigma _1$, $\sigma _2$ and $\sigma _3$; three $1$-dimensional cones $\sigma _4$, $\sigma _5$ and $\sigma _6$; one $0$-dimensional cone $\sigma _0$.

Mathematical diagram from the source
85FCE3A7A73121.13.13 The fan $\Sigma $ of $\mathbb {P}^2$. The three rays $\sigma _4,\sigma _5,\sigma _6\in \Sigma (1)$ subdivide $N_{\mathbb {R}}$ into three full-dimensional cones $\sigma _1,\sigma _2,\sigma _3\in \Sigma (2)$; the origin is the unique cone of dimension zero.

The fan $\Sigma $ is just the fan of $X=\mathbb {P}^2$. Under the orbit-cone correspondence, we have

\[ \begin{aligned} D_{\sigma _1}=& \{ [1:0:0]\} ,\quad D_{\sigma _2}=\{ [0:1:0]\} ,\quad D_{\sigma _3}=\{ [0:0:1]\} ,\\ D_{\sigma _4}=& \{ [0:X_1:X_2]:(X_1,X_2)\neq (0,0)\} ,\\ D_{\sigma _5}=& \{ [X_0:0:X_2]:(X_0,X_2)\neq (0,0)\} ,\\ D_{\sigma _6}=& \{ [X_0:X_1:0]:(X_0,X_1)\neq (0,0)\} ,\quad D_{\sigma _0}=\mathbb {P}^2. \end{aligned} \]

In particular, $\Sigma (1)=\{ \sigma _4,\sigma _5,\sigma _6\} $. We shall take

\[ D=D_{\sigma _4}. \]

In other words,

\[ a_{\sigma _5}=a_{\sigma _6}=0,\quad a_{\sigma _4}=1. \]

Note that the ray generators are given by

\[ u_{\sigma _4}=(-1,-1),\quad u_{\sigma _5}=(1,0),\quad u_{\sigma _6}=(0,1). \]

It follows that

\[ P_ D=\{ m=(m_1,m_2)\in \mathbb {R}^2: m_1+m_2\leq 1, m_1\geq 0, m_2\geq 0\} . \]

Therefore, $P_ D$ is just the polytope in 21.13.14.

Mathematical diagram from the source
16FDB2620E8621.13.14 The polytope $P_ D$.

In this case, the Cartier data for $2$-dimensional cones are given as follows:

\[ m_{\sigma _1}=(0,0),\quad m_{\sigma _2}=(1,0),\quad m_{\sigma _3}=(0,1); \]

while the remaining Cartier data are determined by (21.13.4).

In this case, $L=\mathcal{O}_ X(D)=\mathcal{O}_{\mathbb {P}^2}(1)$. Hence the line bundle $L$ is ample.

We also observe that

\[ \begin{aligned} Q_{\sigma _1}=& \{ (0,0)\} ,\quad Q_{\sigma _2}=\{ (1,0)\} ,\quad Q_{\sigma _3}=\{ (0,1)\} ,\\ Q_{\sigma _4}=& \{ (m_1,m_2):m_1\geq 0,m_2\geq 0,m_1+m_2=1\} ,\\ Q_{\sigma _5}=& \{ 0\} \times [0,1],\quad Q_{\sigma _6}= [0,1]\times \{ 0\} ,\\ Q_{\sigma _0}=& P_ D. \end{aligned} \]
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