1. On the Toric Varieties Associated with Bicolored Metric Trees Khoa Lu Nguyen Joseph Shao Summer REU 2008 at Rutgers University Advisors: Prof. Woodward and Sikimeti Mau

2. Background in Algebraic Geometry An affine variety is a zero set of some polynomials in C[x 1, …, x n ]. Given an ideal I of C[x 1, …, x n ], denote by V(I) the zero set of the polynomials in I. Then V(I) is an affine variety. Given a variety V in C n, denote I(V) to be the set of polynomials in C[x 1, …, x n ] which vanish in V.

3. Background in Algebraic Geometry Define the quotient ring C[V] = C[x 1, …, x n ] / I(V) to be the coordinate ring of the variety V. There is a natural bijective map from the set of maximal ideals of C[V] to the points in the variety V.

4. Background in Algebraic Geometry We can write Spec(C[V]) = V where Spec(C[V]) is the spectrum of the ring C[V], i.e. the set of maximal ideals of C[V].

5. Background in Algebraic Geometry In C n, we define the Zariski topology as follows: a set is closed if and only if it is an affine variety. Example: In C, the closed set in the Zariski topology is a finite set. Open sets in Zariski topology tend to be “large”.

6. Constructing Toric Varieties from Cones A convex cone  in R n is defined to be  = { r 1 v 1 + … + r k v k | r i ≥ 0 } where v 1, …, v n are given vectors in R n Denote the standard dot product in R n. Define the dual cone of  to be the set of linear maps from R n to R such that it is nonnegative in the cone .  v = {u  R n | ≥ 0 for all v  σ}

7. Constructing Toric Varieties from Cones A cone  is rational if its generators v i are in Z n. A cone  is strongly convex if it doesn’t contain a linear subspace. From now on, all the cones  we consider are rational and strongly convex. Denote M = Hom(Z n, Z), i.e. M contains all the linear maps from Z n to Z. View M as a group.

8. Constructing of Toric Varieties from Cones Gordon’s Lemma:  v  M is a finitely generated semigroup. Denote S  =  v  M. Then C[S  ] is a finitely generated commutative C – algebra. Define U  = Spec(C[S  ]) Then U  is an affine variety.

9. Constructing Toric Varieties from Cones Example 1 Consider  to be generated by e 2 and 2e 1 - e 2. Then the dual cone  v is generated by e 1 * and e 1 * + 2 e 2 *. However, the semigroup S  =  v  M is generated by e 1 *, e 1 * + e 2 * and e 1 * + 2 e 2 *.

10. Constructing Toric Varieties from Cones Let x = e 1 *, xy = e 1 * + e 2 *, xy 2 = e 1 * + 2e 2 *. Hence S  = {x a (xy) b (xy 2 ) c = x a+b+c y b+2c | a, b, c  Z  0} Then C[S  ] = C[x, xy, xy 2 ] = C[u, v, w] / (v 2 - uw). Thus U  = { (u,v,w)  C 3 | v 2 – uw = 0 }.

11. Constructing Toric Varieties from Cones If     R n is a face, we have C[S  ]  C[S  ] and hence obtain a natural gluing map U   U . Thus all the U  fit together in U . Remarks Consider the face  = {0}  . Then U  = (C * ) n = T n here C * = C \ {0}. Thus for every cone   R n, there is an embedding T n  U .

12. Constructing Toric Varieties from Cones Example 2 If we denote  0,  1,  2 the faces of the cone . Then U  0 = (C * ) 2  { (u,v,w)  (C*) 3 | v 2 – uw = 0 } U  1 = (C * )  C  { (u,v,w)  (C) 3 | v 2 – uw = 0, u  0} U  2 = (C * )  C  { (u,v,w)  (C) 3 | v 2 – uw = 0, w  0}

13. Action of the Torus From this, we can define the torus T n action on U  as follows: 1. t  T n corresponds to t *  Hom(S {0}, C) 2. x  U  corresponds to x *  Hom(S , C) 3.Thus t *. x *  Hom(S , C) and we denote t. x the corresponding point in S . 4.We have the following group action: T n  U   U  (t, x) |  t. x Torus varieties: contains T n = (C*) n as a dense subset in Zariski topology and has a T n action.

14. Orbits of the Torus Action The toric variety U  is a disjoint union of the orbits of the torus action T n. Given a face     R n. Denote by N   Z n the lattice generated by . Define the quotient lattice N(  ) = Z n / N  and let O  = T N(  ), the torus of the lattice N(  ). Hence O  = T N(  ) =  (C*) k where k is the codimension of  in R n. There is a natural embedding of O  into an orbit of U . Theorem 1 There is a bijective correspondence between orbits and faces. The toric variety U  is a disjoint union of the orbits O  where  are faces of the cones.

15. Torus Varieties Associated With Bicolored Metric Trees Bicolored Metric Trees V(T) = { (x 1, x 2, x 3, x 4, x 5, x 6  C 6 | x 1 x 3 = x 2 x 5, x 3 = x 4, x 5 = x 6 }

16. Torus Varieties Associated With Bicolored Metric Trees Theorem 2 V(T) is an affine toric variety.

17. Description of the Cones for the Toric Varieties Given a bicolored metric tree T. We can decompose the tree T into the sum of smaller bicolored metric trees T 1, …, T m.

18. Description of the Cones for the Toric Varieties We give a description of the cone  (T) by induction on the number of nodes n. For n = 1, we have Thus  (T) is generated by e 1 in C. Suppose we constructed the generators for any trees T with the number of nodes less than n. Decompose the trees into smaller trees T 1, …, T m. For each tree T k, denote by G k the constructed set of generators of  (T k )

19. Description of the Cones for the Toric Varieties Then the set of generators of the cone  (T) is G(T) = {e n +  1 ( z 1 – e n ) + … +  m (z m – e n )| z k  G i,  k  {0,1}} Remarks The cone  (T) is n dimensional, where n is the number of nodes. The elements in G(T) corresponds bijectively to the rays (i.e. the 1- dimensional faces) of the cone  (T)

20. Description of the Cones for the Toric Varieties Example 3 T = T 1 + T 2 + T 3 G(T 1 ) = {e 1 }, G(T 2 ) = {e 2 }, G(T 3 ) = {e 3 } G(T) = {e 1, e 2, e 3, e 1 + e 2 – e 4, e 1 + e 3 – e 4, e 2 + e 3 – e 4, e 1 + e 2 + e 3 - 2e 4 }

21. Weil Divisors We can also list all the toric-invariant prime Weil divisors as follows: Let x 1, …, x N be all the variables we label to the edges of T. We call a subset Y = {y 1, …, y m } of {x 1, …, x N } complete if it has the property that x k = 0 if and only if x k  Y when we set y 1 = … = y m = 0. A complete subset Y is called minimally complete if it doesn’t contain any other complete subset.

22. Weil Divisors A prime Weil divisor D of a variety V is an irreducible subvariety of codimensional 1. A Weil divisor is an integral linear combination of prime Weil divisors. Given a bicolored metric tree T with variety V(T). We care about toric-invariant prime Weil divisors, i.e. T n (D)  D. Theorem 2 D = clos(O  ) for some 1 dimensional face . There is a natural bijective correpondence between the set of prime Weil divisors and the set of generators G(T).

23. Weil Divisors Lemma 1 Y is a minimally complete subset if and only if the unique path from each colored point to the root contains exactly one edge with labelled variable y k. Theorem 3 D is a toric-invariant prime Weil divisor if and only if D ={ (x 1, …, x N )  V(T) | x k = 0,  x k  Y} for some minimally complete Y. Corollary 1 There is a natural bijective correspondence between the set of toric-invariant prime Weil divisors and the set of minimally complete subsets.

24. Weil Divisors We can describe the correpondence map D = clos(O  ) and  by induction on the number of nodes n. Example 4 There are 4 prime Weil divisors D 12, D 156, D 234, D 3456 which correspond to the minimal complete sets Y 12, Y 156, Y 234, Y 3456. Decompose the tree into two trees T 1 and T 2. Then G(T 1 ) = {e 1 } and G(T 2 ) = {e 2 }. Thus D 12, D 156, D 234, D 3456 correspond to e 3, e 2, e 1, e 1 + e 2 - e 3.

25. Cartier Divisors Given any irreducible subvariety D of codimension 1 of V and p  D, we can define ord D, p (f) for every function f  C(V). Informally speaking, ord D, p (f), determines the order of vanishing of f at p along D. It turns out that ord D, p doesn’t depend on p  D. For each f, define div(f) =  D (ord D (f)).D If ord D (f) = 0 for every non toric-invariant prime Weil divisor, we call div(f) a toric-invariant Cartier divisors.

26. Cartier Divisors Theorem 4 div(f) is toric-invariant if and only if f is a fraction of two monomials. Recall that we have constructed a natural correspondence between the prime Weil divisors of V(T) and the elements in G(T). Call D 1, …, D N the prime Weil divisors of V(T) and v 1, …, v N the corresponding elements in G(T). Recall that if the tree T has n nodes, then v 1, …, v N are vectors in Z n.

27. Cartier Divisors Theorem 5 A Weil divisor D =  a i D i is Cartier if and only if there exists a map   Hom(Z n, Z) such that  (v i ) = a i. Example 5 The prime Weil divisors of T are D 12, D 156, D 234, D 3456 corresponds to e 3, e 2, e 1, e 2 + e 3 - e 1. Hence aD 12 + bD 156 + cD 234 + dD 3456 is Cartier if and only if a + d = b + c.

28. Cartier Divisors We can describe the generators of the set of Cartier divisors as follows: 1) Denote the nodes of T to be the point P 1, …, P n. 2) Call Y 1, …, Y N the corresponding minimal complete set of the prime Weil divisors D 1, …, D N. 3) For each node P k, let  k be a map defined on {D 1, …, D N } such that  k (D i ) = 0 if there is no element in Y i in the subtree below P k. Otherwise 1 -  k (D i ) = the number of branches of P k that doesn’t contain an element in Y i.

29. Cartier Divisors Theorem The set of Cartier divisors is generated by {  k (D 1 ) D 1 + … +  k (D N ) D N | 1  k  m} Example: The generators of the Cartier divisors of T are D 12 - D 3456, D 234 + D 3456, D 156 + D 3456.

30. Summary of Results Description of toric-invariant Weil divisors and Cartier divisors of the torus variety associated with a bicolored metric tree.

31. Reference Griffiths, Ph., Harris, J.: Principles of Algebraic Geometry, Wiley, 1978, NY. Miles, R.: Undergraduate Algebraic Geometry, London Mathematical Society Student Texts. 12, Cambridge University Press 1988. Fulton, W.: Introduction to Toric Varieties, Annals of Mathematics Studies 131, Princeton University Press 1993.