# Ample Subvarieties of Algebraic Varieties by Robin Hartshorne, C. Musili

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By Robin Hartshorne, C. Musili

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The Definition of Affine Toric Variety. We now define the main object of study of this chapter. 3. An affine toric variety is an irreducible affine variety V containing a torus TN ≃ (C∗ )n as a Zariski open subset such that the action of TN on itself extends to an algebraic action of TN on V . ) Obvious examples of affine toric varieties are (C∗ )n and Cn . Here are some less trivial examples. 4. The plane curve C = V(x 3 − y 2 ) ⊆ C2 has a cusp at the origin. This is an affine toric variety with torus C \ {0} = C ∩ (C∗ )2 = {(t 2 ,t 3 ) | t ∈ C∗ } ≃ C∗ , where the isomorphism is t → (t 2 ,t 3 ).

If you get stuck, see [204, Thm. 4]. 10. Prove that I = x 2 − 1, xy − 1, yz − 1 is the lattice ideal for the lattice L = {(a, b, c) ∈ Z3 | a + b + c ≡ 0 mod 2} ⊆ Z3 . Also compute the primary decomposition of I to show that I is not prime. 11. Let TN be a torus with character lattice M. Then every point t ∈ TN gives an evaluation map φt : M → C∗ defined by φt (m) = χ m (t). Prove that φt is a group homomorphism and that the map t → φt induces a group isomorphism TN ≃ HomZ (M, C∗ ). 12. Consider tori T1 and T2 with character lattices M1 and M2 .

Prove that dim Tp (Cn ) = n for all p ∈ Cn . 9. 3). Chapter 1. 10. Let V be irreducible and suppose that p ∈ V is smooth. The goal of this exercise is to prove that OV ,p is normal using standard results from commutative algebra. Set n = dim V and consider the ring of formal power series C[[x1 , . . , xn ]]. This is a local ring with maximal ideal m = x1 , . . , xn . We will use three facts: • C[[x1 , . . , xn ]] is a UFD by [280, p. 5. • Since p ∈ V is smooth, [207, §1C] proves the existence of a C-algebra homomorphism OV ,p → C[[x1 , .