Alterations and resolution of singularities

Lecture 3: Toric varieties and toroidal embeddings

3.1. Definition. A variety V together with an open dense embedding T ⊂ V is called a toric variety if V is normal, T is a torus (geometrically isomorphic to 𝔾_{m}^{k}), and the action of T on itself by translations extends to an action on V.

3.2. Definition. A variety X together with an open embedding U ⊂ X is called a toroidal embedding if any point x ∈ X has an étale neighborhood X′ such that X′ ×_{X} U ⊂ U is isomorphic to an étale neighborhood of a point on a toric variety.

Equivalently, given a point x ∈ X, there is a toric variety V, a point v ∈ V, and an isomprphism { \atop O_{X,x}} → { \atop O_{V,v}}, identifying the ideal of X^{\} U with that of V ^{\} T. A toroidal embedding is said to be strict (or a toroidal embedding without self intersections) if every irreducible component of X^{\} U is normal. For instance, if X is a nonsingular variety and D is a strict normal crossings divisor, and U = X^{\} D, then U ⊂ X is a strict toroidal embedding. k = \overline{k} 3.3. Some notation. . M = Hom(T, 𝔾_{m}); M_{ℝ} = M ⊗ ℝ N = Hom(M, ℤ) = Hom(𝔾_{m}, T) ; N_{ℝ} = N ⊗ ℝ Note: M = Monomials. m ∈ M ↦ x^{m}. Recall: 1-1 correspondence

{affine toric varieties T ⊂ V } ↔

{str. conv. rat’l polyhedral cones σ ⊂ N} V_{σ} = Spec k[σ^{∨} ∩ M] σ = cone spanned by the 1-parameter subgroups ϕ : 𝔾_{m} → T such that the limit lim_{z→0} ϕ(z) exists in V.

3.4. Divisorial characterization of σ. Any monomial m ∈ M defines a Cartier divisor Div(x^{m}) supported on V ^{\} T. If σ^{∨} contains a line through the origin, then for any m on this line the divisor is easily seen to be trivial (both m and −m give regular functions). Notation: σ^{⊥} = {m ∈ M_{ℝ}|⟨m, σ⟩ = 0} M^{σ} = Cartier div. supp. on V ^{\} T. One can easily see that M^{σ} = M/σ^{⊥} ∩ M. N_{σ} = span(σ). Clearly N_{σ} = Hom(M^{σ}, ℤ). Let M_{+}^{σ} ⊂ M^{σ}: the effective Cartier divisors. Easy to see: M_{+}^{σ} = σ^{∨} ∩ M/σ^{⊥} ∩ M. Then it is not hard to see that σ = (M_{+}^{σ})_{ℝ}^{∨}, the dual cone of the cone spanned by M_{+}^{σ}. In short: σ is the dual cone to the cone of effective Cartier divisors supported on V ^{\} T.

3.5. The toroidal picture. Let U ⊂ X strict toroidal. X^{\} U = ∪D_{i}, where D_{i} normal. ∩_{i∈I}D_{i} = ∪X_{α}; X_{α} = STRATA Star(X_{α}) = ∪_{X_{α}⊂\overline{X_{β}}}X_{β}. Note: X_{α} = unique closed stratum in Star(X_{α}) M^{α} = Cartier div. supp. on Star(X_{α})^{\} U M_{+}^{α} = Effective Cartier divisors N_{α} = Hom(M_{α}, ℤ) σ_{α} = (M_{+}^{α})_{ℝ}^{∨} Thus: to each stratum we associate a cone.

Remark: σ_{α} has a description analogous to the one using 1-parameter subgroups, in terms of valuations. Let RS(X) be the discrete valuations on X. Let v be a valuation centered in Star(X_{α}). Let f_{j} be rational function defining generators of M_{α} on a small affine open. Then v(f_{j}) is a vector in σ_{α}, and in fact σ_{α} can be described as equivalence classes of discrete valuations centered in Star(X_{α}).

3.6. Birational affine morphisms. Recall: if τ ⊂ σ two s.c.r.p. cones, then τ^{∨} ⊃ σ^{∨} gives rise to a morphism V_{τ} → V_{σ}, which is birational and affine. Note that V_{τ} → V_{σ} can be described in the following invariant manner:

V_{τ} = Spec _{V_{σ}} ∑_{E∈M_{+}^{τ}} O_{V_{σ}}(−E),

where the sum is taken inside the field of rational functions of V_{σ}. This clearly works over Star(X_{α}) in the toroidal case as well.

3.7. Principal affine opens. If m ∈ σ^{∨} then τ = {n ∈ σ| < n, m >= 0} is a face of σ. We have τ^{∨} = σ^{∨} + ℝ · m, and therefore V_{τ} is the principal open set on V_{σ} obtained by inverting the monomial x^{m}. Again, this can be discribed divisorially in terms of Div(m). Thus the same is true for Star(X_{α}): given a face τ of σ_{α}, we get an open set

Star(X_{β}) ⊂ Star(X_{α})

such that τ = σ_{β}. The most important face of a cone is the vertex. It corresponts to the open set T ⊂ V. In the toroidal case you get U.

3.8. Fans. Recall: if σ_{1} and σ_{2} intersect along a common face τ, then V_{σ_{1}} and V_{σ_{2}} can be glued together along the common open set V_{τ}, forming a new toric variety. In general, whenever you have a fan Δ in N, namely a collection of cones σ_{i} intersecting along faces, you can glue together the V_{σ_{i}} and get a toric variety V_{Δ}. It is not hard to see that EVERY toric variety is obtained in this way in a unique manner. The point is that every toric variety is covered by affine open toric varieties. In the toroidal case: X is covered by {Star(X_{α})}_{α}. In general

Star(X_{α}) ∩ Star(X_{β}) = ∪Star(X_{γ_{i}}), so σ_{γ_{i}} are faces of both σ_{α} and σ_{β}. Can glue together, as a rational conical polyhedral complex: abstract, not in some N.

3.9. Modifications and subdivisions. Let Δ be a fan, and Δ′ → Δ a (complete) subdivision. This corresponds to a toric modification V_{Δ′} → V_{Δ}. Since the construction is local (the spec construction and gluing) it works word for word in the toroidal case.

3.10. Nonsingularity. Recall: an affine toric V_{σ} is nonsingular iff σ is simplicial, generated by a basis of N_{σ} (part of a basis of N). Such a cone is called nonsingular. In general: a toric variety V_{Δ} is nonsingular if and only if every cone σ ∈ Δ is nonsingular. This is local, so true for toroidal as well.

3.11. Desingularization. It is easy to resolve singularities of a toric variety: find a simplicial subdivision such that every cone is nonsingular. Obviously now, the same works in toroidal case!