Consider the cone - $\langle e_1,-e_1+2e_2\rangle$ in the lattice $N=\mathbb Z^2$. Then it has the following faces -

  1. $0=\{(0,0)\}$

  2. $\rho_1=\langle e_1\rangle$

  3. $\rho_2=\langle-e_1+2e_2\rangle$

  4. $\sigma=\langle e_1,-e_1+2e_2\rangle$

I am trying to compute the distinguished points corresponding to each face.

We know that corresponding to a cone $\sigma$ we have an affine toric variety $U_\sigma=\text{Hom}_{sg}(\sigma^\vee\cap M,\mathbb C)\cong \text{MaxSpec }\mathbb C[\sigma^\vee\cap M]$. Where $M$ is the dual lattice.

Corresponding to each face $\tau$ of $\sigma$ we define the distinguished point in $\gamma_\sigma \in U_\sigma$ as $$\gamma_\sigma(m)=\left\{\begin{array}[rcl]0&1&m\in\tau^\perp\cap M\\ &0& \text{otherwise}\end{array}\right.$$

I am trying to find $\gamma_0$ , $\gamma_{\rho_1}$, $\gamma_{\rho_2}$ and $\gamma_\sigma$. I know the following result -

Theorem - If $u\in\text{Relint }\sigma$ then $\lim_{t\to0}\lambda^u(t)=\gamma_\sigma$.

Where $\lambda^u$ is the one parameter subgroup corresponding to $u$. In this case, for any $u=(a,b)\in N$, $\lambda^u(t):=(t^a,t^b)$.

Using the above result I have got that $\gamma_0=(1,1)$ , $\gamma_{\rho_1}=(0,1)$, $\gamma_{\rho_2}=(0,0)$.

  1. Is what I have done so far correct?

  2. I am stuck with $\gamma_\sigma$. If I take $u=e_2$ then I get the limit as $(1,0)$ but if I take $u=ae_1+be_2$ , $a,b\ge0$ I get $(0,0)$ and if I take $u=a(-e_1+2e_2)-b_2$, $a,b\ge0$ then the limit doesn't exist.

What am I doing wrong?

Thank you.



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