for counting plane curves

Transcription

1 for counting plane curves Hannah Markwig joint with Andreas Gathmann and Michael Kerber Georg-August-Universität Göttingen Barcelona, September 2008

2 General introduction to tropical geometry and curve-counting

3 Idea of tropical geometry Replace algebraic varieties by degenerations, tropical varieties, which are piece-wise linear objects. The tropical varieties can be studied using combinatorics and linear algebra methods. Use the tropical varieties to prove theorems about algebraic varieties.

4 dual Newton subdivisions

5 more dual Newton subdivisions

6 Tropical fans The gluing pieces for tropical varieties. Λ denotes a lattice (Z N ) and V = Λ Z R the corresponding real vector space. Definition A tropical fan in V is a fan such that 1 cones σ are cut out by integral inequalities and equations (Λ σ is the smallest sublattice of Λ that contains σ Λ), 2 full-dimensional cones σ are equipped with a weight ω(σ) N >0, and 3 for each cone τ of codimension 1 the balancing condition is fulfilled.

7 For a fan of dimension 1 the balancing condition means: ω = 2 ` 1 0 `1 1 ` 1 1 For a higher-dimensional fan, project the cone τ and its full-dimensional neighboring cones along τ:

8 Definition Let N d denote the number of nodal rational plane curves of degree d passing through 3d 1 points in general position. N 1 = 1 N 2 = 1 N 3 = 12 Theorem (Kontsevich) N d = d 1+d 2=d d 1,d 2>0 ( ( 3d 4 d 2 1d 2 2 3d 1 2 ) ( )) 3d 4 d 3 1d 2 N d1 N d2 3d 1 1

9 counting Definition Let N trop d denote the number of tropical rational plane curves of degree d passing through 3d 1 points in general position. Theorem (Mikhalkin s Correspondence Theorem) N d = N trop d have to be counted with multiplicity mult = number of algebraic curves that degenerate to the tropical curve multiplicity can be defined combinatorially

10 a tropical count of N 3 4

11 Some tropical,,

12 Tropical fans Λ denotes a lattice (Z N ) and V = Λ Z R the corresponding real vector space. Definition A tropical fan in V is a fan such that 1 cones σ are cut out by integral inequalities and equations (Λ σ is the smallest sublattice of Λ that contains σ Λ), 2 full-dimensional cones σ are equipped with a weight ω(σ) N >0, and 3 for each cone τ of codimension 1 the balancing condition is fulfilled.

13 For a fan of dimension 1 the balancing condition means: ω = 2 ` 1 0 `1 1 ` 1 1 For a higher-dimensional fan, project the cone τ and its full-dimensional neighboring cones along τ:

14 Two are called equivalent, if they have a common refinement. (The weights have to agree, too.) A tropical fan is called irreducible, if there is no tropical fan of the same dimension which is strictly contained.

15 Lemma Let X and Y be of dimension n, X irreducible, and Y contained in X. Then Y = λ X for some λ in Q >0. Idea of the proof: suitable refinement λ := min σ X ω Y (σ)/ω X (σ) α such that α λ Z new weight function: ω(σ) = α (ω Y (σ) λω X (σ)) new fan containing only σ ω(σ) > 0. It has to be empty.

16 Definition A morphism of fans f : X Y is a Z-linear map, that is, a map induced by a linear map from Λ to Λ. construct the image fan f(x) in Y : roughly: cones f(σ), where σ is contained in a full-dimensional cone on which f is injective. the cones f(σ) might overlap suitable refinement define the weight of σ in f(x) to be ω f(x) (σ ) = σ X f(σ)=σ ω X (σ) Λ σ /f(λ σ). f(x) is a tropical fan, too.

17 Corollary X, Y of the same dimension n, f : X Y. Y irreducible. For Q such that Q σ Q, dim(σ Q ) = n inverse images P σ P, dim(σ P ) = n define the multiplicity Then the sum mult P f := ω X(σ P ) ω Y (σ Q ) Λ σ Q /f(λ σ P ). P f(p )=Q mult P f does not depend on Q( degree of f ).

18 Idea of the proof: suitable refinements Y irreducible = f(x) = λ Y for some λ. Therefore P f(p )=Q = ω f(x)(σ ) ω Y (σ ) mult P f = = λ does not depend on Q. σ f(σ)=σ ω X (σ) ω Y (σ ) Λ σ /f(λ σ)

19 Tropical M trop,0,n Definition An abstract tropical curve with n markings is a tree, such that each vertex is at least 3-valent, n of the leaves (ends, unbounded edges) are marked and the bounded edges are equipped with a (positive) length. M trop,0,n = space of abst. trop. curves with N ends, all marked x 2 l 1 = 4 x 1 l 2 = 2.5 x 3 x 4

20 M trop,0,n can be embedded as a tropical fan in R (N 2) N : ij-th coordinate given by the distance of leaf i and leaf j Divide out W spanned by trees with a 2-valent and an N-valent vertex:

21 M trop,0,4 can be embedded into R 2 = R 6 /W. It consists of three rays generated by (0, 1, 1, 1, 1, 0) (1, 0, 1, 1, 0, 1) (1, 1, 0, 0, 1, 1) Their sum = the sum of the 4 generators of W = M trop,0,4 is a tropical fan.

22 Definition Parametrized (Γ, x i, h) is an n-marked parametrized tropical curve of degree d to R 2 if: (Γ, x i ) abstract tropical curve with N = n + 3d ends, h : Γ R 2 a continuous map satisfying: On each edge E, h is of the form h E : [0, l(e)] R 2 : t a + v(e) t (a R 2, v(e) Z 2 direction of E). (v(e) = product of the weight ω(e) and the primitive integral vector.) For every vertex the balancing condition holds. Marked ends x i contracted to a point by h (i.e. v(x i ) = 0). d of the other ends map to ( 1, 0), d to (0, 1), d to (1, 1).

23 The length of an image edge h(e) is determined by the length of E in the abstract tropical curve and the direction v(e) (if v(e) 0). The space of all n-marked parametrized tropical curve of degree d to R 2 is denoted by M trop,0,n (R 2, d).

24 How can we make M trop,0,n (R 2, d) a trop fan? Use M trop,0,n! Mark the other ends by labels y i. All directions are determined by the directions of the ends: = the map h is determined by the position of one point (e.g. h(x 1 )) in R 2, the abstract curve (Γ, x i, y i ) and the directions v(y i ).

25 Get a map M trop,0,n R 2 M trop,0,n (R 2, d). Cover, number of inverse images varies for the different cones. Example: M trop,0,0 (R 2, 2). The picture encodes both graph and map. Avoid this by labelling the ends (for enumerative statements, we have to divide by (d!) 3 later.)

26 in the sense: M lab trop,0,n (R2, d) = M trop,0,n R 2 (Γ, x i, y i, h) ((Γ, x i, y i ), h(x 1 )) bijection between the two sets of cones, each cone is equal to its image cone under this bijection, gluing coincides. The right hand side is a tropical fan.

27 Definition The map ev i : M lab trop,0,n(r 2, d) R 2 (Γ, x 1,... x n, y i, h) h(x i ) is called the i-th evaluation map. Lemma The i-th evaluation map ev i is a morphism of fans.

28 Idea of the proof: Identify M lab 0,n,trop(R 2, d) with M 0,N,trop R 2. Define ev i ev i : R (N 2) R 2 R 2 (a 1,2,... a N 1,N, b) b is 0 on W and it is a linear map. I x 1 l(e) = 1 v(e) N (a 1,k a i,k ) v k k=1 x i J For all ends k I, a 1,k = 0, a i,k = 1 = count negatively. For all ends l J, a 1,l = 1, a i,l = 0 = count positively. Balancing condition = k I v k = l J v l = v(e).

29 ft n : M 0,n,trop M 0,n 1,trop take the subgraph not containing end n and straighten 2-valent vertices l 1 l 2 n l 1 + l 2

30 Conclusion Now we know: moduli of are evaluation maps and are tropical morphisms degree of tropical morphism is constant. We will use this in the next talk to derive Kontsevich s!

31 Kontsevich s tropically

32 Definition Parametrized (Γ, x i, h) is an n-marked parametrized tropical curve of degree d to R 2 if: (Γ, x i ) abstract tropical curve with N = n + 3d ends, h : Γ R 2 a continuous map satisfying: On each edge E, h is of the form h E : [0, l(e)] R 2 : t a + v(e) t (a R 2, v(e) Z 2 direction of E). (v(e) = product of the weight ω(e) and the primitive integral vector.) For every vertex the balancing condition holds. Marked ends x i contracted to a point by h (i.e. v(x i ) = 0). d of the other ends map to ( 1, 0), d to (0, 1), d to (1, 1).

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34 the forgetful map to M 0,4,trop ft : M 0,n,trop (R 2, d) M 0,4,trop x 1 x 2 x 3 l h 1 x 5 l 2 x 4 x 1 x 3 l 1 + l 2 R 2 h(x 1) 14/23 l 1 + l 2 x 2 12/34 x 4 13/24

35 Pick a configuration P of a vertical line L 1 a horizontal line L 2 points p 1,..., p 3d in the plane a point λ M 0,4,trop. Count curves passing through the lines and points and mapping to λ under ft. Definition π := ev 1 1 ev 2 2 ev 3... ev 3d ft : M 0,3d,trop (R 2, d) R R (R 2 ) 3d 2 M 0,4,trop

36 contracted bounded edge Forget the λ-condition. Curves through L 1, L 2, p 3,..., p 3d. A 1-dim family. The movement is bounded:

37 local picture of a contracted bounded edge: v 0 w h global picture: v w h R 2 The curves become reducible!

38 N trop d = d 1+d 2=d d 1,d 2>0 ( d 2 1 d2 2 ( ) 3d 4 d 3 1 3d 1 2 d 2 ( )) 3d 4 3d 1 1 N trop d 1 N trop d 2

39 References: A. Gathmann, H. Markwig, Kontsevich s and the WDVV equations in tropical geometry, Advances in Mathematics (to appear). A. Gathmann, M. Kerber, H. Markwig, Tropical fans and the moduli space of rational, Compositio Mathematica (to appear). G. Mikhalkin, Enumerative tropical geometry in R 2, J. Amer. Math. Soc. (18) 2005, G. Mikhalkin, Tropical Geometry and its applications, Proceedings of the ICM, Madrid Spain (2006), THANK YOU!