let $\tau, \sigma \subseteq N_{\mathbb{R}}$ be two rational, strongly convex polyhedral cones with $\tau \subseteq \sigma$. Now we get an inclusion $S_{\sigma} \to S_{\tau}$ inducing an inclusion $\mathbb{C}[S_{\sigma}] \to \mathbb{C}[S_{\tau}]$ and thus a morphism of varieties $U_{\tau} \to U_{\sigma}$. Assume that this is an open embedding.
Question: Why is $\tau$ then a face of $\sigma$?
This is exercise $3.2.10$ in Cox, Little and Schenck and I don't know how to prove it. Does this have anything to do with torus orbits? How exactly can we use that the induced mapping is an open embedding?
In the exercise they say I should prove it like that: If $u, u' \in N \cap \sigma$ and $u + u' \in \tau$ one has to show that $u \in \tau$ and $u' \in \tau$. Now they look at the limit points corresponding to the one-parameter subgroups and I understand, why $\lim_{t \to 0} \lambda^u(t) \cdot \lim_{t \to 0} \lambda^{u'}(t)$ (which lies in $U_{\sigma}$, as $u, u' \in \sigma$) lies in $U_{\tau}$ (which follows since $u + u' \in \tau$). How can i deduce that each of the limits already lies in $U_{\tau}$?
Thanks for all answers.