Past exam of the mathematics course of the University of Cambridge / 2024 / iii / Paper 152 / 1 / b

Table of contents
- Solution b

Solution

The orbit-cone correspondence is an inclusion-reversing bijection

{σ \in Σ} ⟷ {T_{N} -orbits in X_{Σ}}, σ ⟼ O (σ) .

If

n = rank N

, then

dim O (σ) = n - dim σ, O (σ) ≅ Hom (M \cap σ^{⊥}, C^{\times}) .

The affine chart and orbit closure in a toric variety are

U_{σ} = ⋃_{τ ⪯ σ} O (τ), V (σ) = \overline{O (σ)} = ⋃_{τ \supseteq σ} O (τ) .

In particular, the zero cone corresponds to the dense algebraic torus, while maximal cones correspond to torus-fixed points.

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