The <Cox construction> uses coordinates $x_1,\ldots,x_4$ of degrees
$$
\deg x_1=\deg x_3=F,
\qquad
\deg x_2=S,
\qquad
\deg x_4=S+kF.
$$
Its irrelevant locus is
$$
V(x_1,x_3)\cup V(x_2,x_4),
$$
so put $U=\mathbb A^4\setminus\bigl(V(x_1,x_3)\cup V(x_2,x_4)\bigr)$. The <algebraic torus> $G=(\mathbb G_m)^2$ acts by
$$
(\lambda,\mu)\mathbin{\cdot}(x_1,x_2,x_3,x_4)
=(\mu x_1,\lambda x_2,\mu x_3,\lambda\mu^k x_4).
$$
Every point of $U$ has a nonzero coordinate from each opposing pair, so the action has the expected closed orbits and trivial stabilizers. The <geometric quotient> is
$$
\mathbb F_k=U/G.
$$

Solved by gpt-5.6-sol high.

Solution

Past exam of the mathematics course of the University of Cambridge / 2026 / iii / Paper 152 / 2 / b / Solution

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Toric geometry studies algebraic varieties containing an <algebraic torus> as a dense open subset, with the torus action extended to the whole variety. Rational polyhedral fans translate their geometry into lattice combinatorics.

Toric geometry

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A toric variety $X_\Sigma$ is obtained by gluing the affine toric varieties associated with the cones of a fan $\Sigma$. It contains an <algebraic torus> as a dense open orbit.

Toric variety

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An algebraic torus of dimension $n$ is an algebraic group isomorphic over the base field to $(\mathbb G_m)^n$. Its character and cocharacter lattices are dual free abelian groups $M$ and $N$.


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 Incoming links: Algebraic torus