Part III - Toric Geometry Lectured by Dhruv Ranganathan Comments or corrections should be sent to [email protected]. Last updated: October 2020 Contents 1 Introduction 2 2 Fans, cones, and toric varieties over C 5 2.1 Convex geometry . .6 2.2 Gluing affines . .7 3 Properties of XΣ 10 3.1 Smoothness . 10 3.2 Normality . 11 3.3 Properness . 11 3.4 Polytope construction . 14 4 Morphisms 15 4.1 Blowups . 16 4.2 Resolution of singularities . 17 5 Orbit-cone correspondence 20 5.1 Orbit corresponding to a cone . 20 5.2 Orbit closures . 22 5.3 Examples . 23 6 Divisors 24 6.1 Definitions . 24 6.2 Weil divisors on toric varieties . 26 6.3 Cartier divisors on toric varieties . 28 6.4 Introduction to intersection theory . 31 7 Line bundles 32 7.1 Polytopes and global sections of divisors . 32 7.2 Motivation: projective morphisms . 34 7.3 Positivity vs convexity . 35 7.4 Non-projective proper varieties . 36 8 Further topics 38 8.1 Semi-stable reduction . 38 8.2 Chow groups and ring structure . 41 8.3 Logarithmic geometry . 43 Appendices 45 A Log geometry 45 B Cox ring and “quotients” 47 C Tropical geometry 49 1 1 Introduction In general, algebraic geometry deals with this duality between fpolynomial equationsg fvarieties (schemes)g f V(f). ! Many times a statement that is hard to prove on one ! side will be manageable on the other side, and we try to understand things from the knowledge of both sides. Let’s consider the ring S = C[x1, ::: , xn], and I an ideal of S, (finitely) generated by f1, ::: , fs. In n α a1 α2 αn what is to come, we use multi-index notation: for α 2 N , write x = x1 x2 ··· xn . There are three flavors of polynomial theories depending on the types of fi: α • Monomials: each fi has only one term (i.e. fi = cix ). A monomial ideal is an ideal I ⊂ S (finitely) generated by monomials. If I is monomial, then V(I) is a union of coordinate subspaces of Cn. Examples: 1. x1 = 0 in S. 2. x1x2 = 0 in C[x1, x2]. The cross is called “reducible” as a union of two (irreducible) lines. 2 2 3. x1 = 0. Then V(x1) is the x2 axis. But this is not “reduced” in the sense that x1 is nilpotent. • Let’s come back to binomials in a second. • Trinomials: each fi has at most 3 non-zero monomial terms. E.g. x1 + x2 + x3 is a trinomial, but 2 4 8 x1 + x2 + x3 + x4 isn’t. Lemma. Any affine variety is a vanishing locus of a trinomial ideal. More precisely, we are saying S/I as a ring is isomorphic to C[x1. ::: , xN]/a for some (usually larger) N, where a is generated by trinomials. Although this is a nice result, it is pretty useless, and nothing is important here. α α Proof. (sketch) For any polynomial a1x 1 + ··· + alx l in I, we introduce l - 3 new variables l-3 α1 αl fzigi=1. Then we can replace a1x + ··· + alx = 0 by α1 α2 α3 αl-1 αl a1x + a2x + z1 = -z1 + z2 + a3x = ··· = -zl-3 + al-1x + alx = 0. Repeat this for every equation and we are done. Note this is a much larger ambient space. • Binomials: let I be generated by binomials. Assume here I is prime (then V(I) will be both irreducible and reduced. Call this “integral”). Proposition. Let C ⊃ X = V(I) for the above I, with dim X = m. Then X contains (a copy of ) (C∗)m as a Zariski dense open subset. Moreover, the multiplication map (group operation): (C∗)m × (C∗)m (C∗)m extends to a map ! X × (C∗)m X. ! 2 Examples: 1. Cn with I = (0). This obviously contains (C∗)n as a dense open. The map is defined by Cn × (C∗)n Cn (z1, ::: , zn) × (t1, ::: , tn) 7 (z1t1, ::: , zntn). ! 3 2 2. Take the hyperplane V(x1 - x2) in C , which is isomorphic! to C (and is 2-dimensional). It ∗ 2 0 0 ∗ contains (C ) = f(t, t, t ) j t, t 2 C g. It extends to the whole V(x1 - x2) as 0 times anything is 0. 2 3 ∗ 2 2 0 0 3. Take V(x1 - x2) in C , which is again 2-dimensional. It contains (C ) = f(t, t , t ) j t, t 2 C∗g. u Key fact: monomial is a homomorphism (C∗)n - C∗. So the ideal I encodes (as we require binomials equal 0) the intersection of translates of the kernels of the homomorphisms u. Thus there is a hidden algebraic torus structure here. ! ∗ Definition. An algebraic torus over C of rank 1 is a group variety Gm = C . A rank n algebraic torus is n ∗ n (Gm) = (C ) . By group variety (aka algebraic group), we are saying: • There are usual group axioms on Gm. ∗ n • It’s a variety: (C ) = V(x1x2 ··· xn+1 - 1) (RHS mapping to (x1, ::: , xn) gives the required identity). • Both: the multiplication and its inverse are both morphisms of varieties. lecture 2 If X is an affine variety, X = V(I) n for some ideal I ⊂ C[x1, ::: , xn]. But by doing so, we are embedding X into the ambient space AC . S/I has many presentation of a quotient of an polynomial ring. Notation: Spec(S/I) = V(I), and Spec(R) = V(I) where R = S/I. Definition. An (affine) toric variety X is an irreducible (affine) variety over C of dimension l, containing a copy of (C∗)l as a dense open subset, such that the multiplication map m :(C∗)l × (C∗)l (C∗)l extends to a group action X × (C∗)l X. ! Remark. ! 1. Since X is irreducible, Zariski open & non-empty imply dense. 2. We can drop affine, and by variety, we mean a finite-type C-scheme. ∗ l 1 3. The embedding of (C ) is part of the data. Any p1 6= p2 on P can be moved to 0 and by a 1 ∼ ∗ Mobius map, and thus P n fp1, p2g = C . 1 Example. 1. (C∗)l and Cl. 2. (C∗)l1 × (C∗)l2 . 3 l 3. P . Pick homogeneous coordinates x0, ::: , xl. Then identify ∗ l ∗ l (C ) = f[1 : t1 : ··· : tl] j ti 2 C g , P . To check density and openness, note: ! l l ∗ l P ⊃ C = f[x0 : ··· : xl] j x0 6= 0g ⊃ (C ) = fall xi 6= 0g. 4. Pl1 × · · · × Plk . Remark. There are infinitely many toric varieties in every dimension l ≥ 2. What is more interesting is that there exist toric varieties X where • X is singular. • X is compact but not projective. • Pic(X) has torsion. All such cases can be constructed using combinatorics. Two lattices we will use frequently: ∗ ∗ l ∼ l • N = Homalg grp(C , (C ) ) = Z , called the cocharacter lattice, where the isomorphism is given by (RHS to LHS): λ1 λl (λ1, ::: , λl) 7 t 7 (t , ::: , t ) . ∗ l ∗ ∼ l • M = Homalg grp((C ) , C ) = Z , called the!character! lattice, where the isomorphism is given by : u1 ul (u1, ::: , ul) 7 (t1, ::: , tl) 7 t1 ··· tl . l ! ! Let X be a toric variety with torus Gm and lattices N and M. Given λ 2 N in the cocharacter lattice, i.e. a morphism λ(t): C∗ (C∗)l ⊂ X, λ(t) we can ask: does limt exist? If so, what is! it? 2 ∗ 2 2 Example. P with (C!1) = f[1 : t1 : t2]g, and N = Z . 2 2 Limit always exists, as X = P is proper. But what is the limit? Let λ = (λ1, λ2) 2 Z . Let’s consider the case when t 0. 1. λ1, λ2 > 0.! Then λ : C∗ (C∗)2 t 7 [1 : tλ1 : tλ2 ] [1 : 0 : 0]. ! ! ! 2. λ1 = 0 and λ2 > 0. t 7 [1 : 1 : tλ2 ] [1 : 1 : 0]. 3. λ1 > λ2 and λ2 < 0. ! ! [1 : tλ1 : tλ2 ] = [t-λ2 : tλ1-λ2 : 1] [0 : 0 : 1]. etc. ! 4 2 There are a total of 7 cases, making a fan in N ⊗ R = NR = R of the following form (3 areas, 3 axises, and origin): More pictures: P1: in R P1 × P1: Every toric variety X determines such a fan Σ ⊂ NR. By fan, we mean a union of cones in NR, where a cone is of the form cone(S) = fv 2 NR j r i=1 νiλi, νi ≥ 0g for a finite set N ⊃ S = fλ1, ::: , λrg. We will make this precise next time. P 2 Fans, cones, and toric varieties over C e lcue3 Letlecture X be a toric variety of dimension n, i.e. ∗ n ∼ n X ⊃ (C ) = Gm. ∗ ∗ l ∼ l The cocharacter lattice N = Homalg grp(C , (C ) ) = Z . 2 Last time: for P we saw that, by examining each λ 2 N and the values of limt 0 λ(t) 2 X, we obtained a diagram: ! Fact: this construction works for any toric variety X, and will give rise to ΣX a collection of subsets of N. n n Example. X = C. Then ΣX = ΣCn is R≥0 ⊂ R . If n = 2, we get 4 subsets (1 area, 2 axes and origin): and outside this first quadrant, the limit does not exist. Natural questions to ask: what property does ΣX have? Does this construction X ΣX have an inverse construction? 5 2.1 Convex geometry Let N, M be as before, the cocharacter and character lattices. By definition, there is a pairing h·, ·i : N × M Z. Definition.! A cone σ in NR is the non-negative span of a finite set S ⊂ N.