Institutionen för matematik, KTH.

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1 Institutionen för matematik, KTH.

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3 Chapter 11 Construction of toric varieties 11.1 Recap example Example Consider the Segre embedding seg : P 1 P 1 P 3 given by ((x 0, x 1 ), (y 0, y 1 )) (x 0 y 0, x 0 y 1, x 1 y 0, x 1 y 1 ). Consider now the map φ : (C ) 2 (C ) 4 given by φ(t 1, t 2 ) = (1, t 1, t 2, t 1 t 2 ). Observe that if one identifies (C) 2 with the Zariski open P 1 P 1 \ (V (x 0 ) V (y 0 )) then it is φ = seg (C ) 2. This image is a torus which shows that the torus (C ) 2 can be identified with a Zariski open of the Segre variety Im(seg) P 3. The torus action of (C ) 2 on Im(seg) defined by (t 1, t 2 ) (x 0, x 1, x 3, x 4 ) = (x 0, t 1 x 1, t 2 x 3, t 1 t 2 x 4 ) is by definition an extension of the multiplicative self-action. Notice that in Example the map defining the toric embedding and the torus action was given by characters associated to the vertices of the polytope 1 1. Observe moreover that for this polytope the vertices coincide with all the lattice points in the polytope. This is of course not always the case, the polytope 2 2 for example is the convex hull of 3 vertices, but it contains 2 2 Z 2 = 6 lattice points. Example Let A = 2 2 Z 2 = {(0, 0), (0, 1), (1, 0), (1, 1)(0, 2), (2, 0)}. Consider the map defined by the associated characters and the following composition: φ A : (C ) 2 C 6 P 5, (t 1, t 2 ) (1, t 1, t 2, t 1 t 2, t 2 1, t 2 2). Observe that this map is the restriction of the 2-Veronese embedding. One sees as above that such a variety is a two dimensional projective toric variety. The previous examples suggest a general construction: 1

4 11.2 Toric varieties from polytopes Let T be an n-dimensional torus with character group M = Z n A = {m 0,..., m d } M. Consider the following action of T on C d+1 and let t (x 0,..., x d ) = (χ m 0 (t)x 0,..., χ m d (t)x d ). This action yields an action on the projective space P d as t (λx 0,..., λx d ) = λ(χ m 0 (t)x 0,..., χ m d (t)xd ). Let x 0 P d be a general points, i.e. a points with non-zero homogeneous coordinates. The orbit T x 0 = T A = T. The Zarisky closure in P d of the orbit x 0 = T is a projective algebraic variety containing a torus as Zarisky open set. Let X A = T A to be such variety. Alternatively: Let P R n be an n-dimensional polytope and let A = P Z n = {m 0,..., m d }. Assume that m 0 = 0 and that P A is contained in the positive orthant. Consider the monomial map defined by the associated characters: φ A : (C ) n C d+1 P d, (t 1,..., t n ) = t (1 : t m 1 :... : t m d ) The image Im(φ A ) is a torus T A. Define X A to be the Zariski closure of T A. This means that X A is the smallest subvariety of P d containing T A. Let A denote the n (d + 1) matrix whose columns are the vectors m i. Lemma Th variety X A is a projective toric variety of dimension equal to rank(a). Proof. Let T A = (C ) r and consider the lattice of its characters: Hom AG (T A, C ) = Z r. The map φ A induces a map: Hom AG ((C ) d+1, C ) Hom AG ((C ) n, C ); f f φ A ψ A : Z d+1 Z n, e i m i where e i are the elements of the standard lattice basis. We see that ψ A (Z d+1 ) = Z r, and thus that r = rank(a). Excercise Consider the n dimensional standard simplex n = Conv(e 0, e 1,..., e n ), where e 0 = 0. Describe the projective toric variety associated to n and 2 n. Let P R n be an n-dimensional lattice polytope. The toric variety associated to P, denoted by X P is the topic variety X P Z n. 2

5 11.3 Affine patching and subvarieties 11.4 Recap example You have seen that P n is the projective toric variety associated to the polytope n. By translating any vertex e i to e 0 = 0 one can contruct a map: φ i : (C ) n C n P n defined by t (t e 0 e i,..., 1, t en e i ). The Zariski closure of Im(φ i ) defines the affine patch of P n where x i 0, i.e. Im(φ i ) = X i Notice that the map φ i is the map defined by the lattice points: A i = {e 0 e i, e i e i,..., e n e i } We will see that projective toric varieties are in a sense a generalisation of projective space as they are built by patching together affine toric varieties defined by the vertices of the polytope Affine patching Let P R n be a polytope and let A = P Z n = {m 0,..., m d }. For every m i A define A mi = {m m i m A}. Consider φ Am : (C ) n C d, t (..., t m j m i,...) mj A and define: Not that X m is an affine toric variety. X m = Im(φ Am ) C d. Proposition Notation as above. Let V = {v 1,..., v r } be the set of vertices of P. Then X A = X vi. v i V Proof. First notice that X mi = X A X i P d and thus X A = m A X m. We prove the proposition if we show that or every m A there is at least one vertex v V such that X m X v. As observed P = Conv(V ). Let m = v i V k iv i. After clearing denominators we can write km = v i V k iv i, for k i Z 0. Notice that t m 0 iff t km = t k i v i = Π(t v i ) k i 0, which happens only if t vi 0 for every k i 0. This shows that X m X vi for every k i 0. 3

6 The vertices of the polytope defines the affine patches that bild the associated toric variety. The following gives an intuition of how projective toric varieties are considered a generalisation of the projective space. Excercise Let P be a polytope of dimension n and let P Z n = {m 0,..., m d }. Show that d n d = n and m 1,..., m n is a lattice basis (i.e. every vector in Z n is an integral combination of m 1,..., m n ) if and only if P = n. Let us now examine closer the category of smooth polytopes and the associated topic varieties. Let P be a smooth polytope anklet m 0 be a vertex. After a lattice-preserving affine transformation can we assume that m 0 = 0 and that the primitive vectors on the n edges through m 0 are e 1,..., e n. Lemma (Exercise) Let P be a smooth polytope. Then X v = C n for every vertex v. Observe that if P is a n-dimensional smooth lattice polytope, then a facet F P is a smooth polytopes of dimension (n 1). Denote by X F the associated topic variety. Lemma Let P be a smooth polytope. Then X P \ T P = F facet X F. Proof. Let dim(p ) = n, let V denote the set of vertices of P and V (F ) denote the set of vertices of F. First observe that: X P \ T P = v V (X v \ T P ) = v V ( i ({(x 1,..., x n ) X v s.t. x i = 0})). Let v = (m 1,..., m n ) V, then are n facets passing through v, F 1,..., F n such that v i = (m i,..., m i 1, m i+1, m n ) V (F i ). Clearly it is: {(x 1,..., x n ) X v s.t. x i = 0} = X vi X Fi. This proves that X P \ T P F facet X F. But because for each facet it is X F = w V (F ) X w and w = v i for some v V, it is clearly X F vi =w,w V (F )X v \ T N and thus F facet X F X P \ T P. 4

7 11.6 Assignment: exercises (1) Prove Lemma (2) Recall that kp = {m m k s.t. m i P } and that if P 1 R n, P 2 R t then P 1 P 2 = {(m, n) s.t. m P 1, m P 2 } R n R t is a polytope of dimension dim(p 1 ) + dim(p 2 ) and whose faces are products of faces of resp. polytopes. (a) Describe the faces of the polytope P = (b) Is P smooth? (c) Describe the toric variety X P as union of affine patches. (d) Describe the induced map X P P 11. 5