For $m=(u,v)\in M\cong\mathbb Z^2$, membership in the <dual cone> is equivalent to
$$
u\geq0,
\qquad -u+3v\geq0.
$$
The <Hilbert basis of a rational cone> is $(0,1),(1,1),(2,1),(3,1)$. Therefore the <coordinate ring of an affine toric variety> is
$$
\mathbb C[S_\sigma]
=\mathbb C[t_2,t_1t_2,t_1^2t_2,t_1^3t_2]
\subseteq\mathbb C[t_1^{\pm1},t_2^{\pm1}].
$$

Solution

Past exam of the mathematics course of the University of Cambridge / 2022 / iii / Paper 152 / 1 / a / i / Solution

 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2022/iii/paper-152/1/a/i/solution