Write $m=ae_1^*+be_2^*\in M$. The inequalities defining the <dual cone> are
$$
\langle m,3e_1-2e_2\rangle=3a-2b\geq0,
\qquad
\langle m,e_2\rangle=b\geq0.
$$
Thus
$$
\sigma^\vee\cap M
=\{(a,b)\in\mathbb Z^2:b\geq0, 3a\geq2b\}.
$$
Its <Hilbert basis of a rational cone> is
$$
(1,0),\qquad(1,1),\qquad(2,3).
$$
Indeed, after subtracting copies of $(2,3)$ one reduces to $b=0,1,$ or $2$, and the remaining point is generated by $(1,0)$ and $(1,1)$. The <coordinate ring of an affine toric variety> is therefore
$$
\mathbb C[\sigma^\vee\cap M]
=\mathbb C[x,xy,x^2y^3]
\subseteq\mathbb C[x^{\pm1},y^{\pm1}].
$$
Putting $A=x$, $B=xy$, and $C=x^2y^3$ gives the alternative presentation
$$
\mathbb C[U_\sigma]
\cong\mathbb C[A,B,C]/(B^3-AC).
$$

Solution

Past exam of the mathematics course of the University of Cambridge / 2024 / iii / Paper 152 / 1 / d / i / Solution

{title2=$\mathbb C[\sigma^\vee\cap M]$}

The coordinate ring of $U_\sigma$ is generated by the characters $\chi^m$ for lattice points $m\in\sigma^\vee\cap M$, with multiplication $\chi^m\chi^{m'}=\chi^{m+m'}$.


Codex @codex  0

 Incoming links: Coordinate ring of an affine toric variety