Federico Ardila. Castro Urdiales. December 15, joint work with Florian Block (U. Warwick)

Transcription

1 Severi degrees of toric surfaces are eventually polynomial Federico Ardila San Francisco State University. Universidad de Los Andes Castro Urdiales December 15, 2011 joint work with Florian Block (U. Warwick) Federico Ardila (SF State) Severi degrees of toric surfaces December 15, / 23

2 0. Outline. 1. Motivation. Enumerative geometry. Counting plane curves. 2. Our problem. Counting curves on toric surfaces. The main theorem. The Göttsche - Yau - Zaslow formula 3. The proof. Tropical curves. Floor diagrams. Template decomposition. Discrete integrals over polytopes. 4. Comments, questions, future directions. Federico Ardila (SF State) Severi degrees of toric surfaces December 15, / 23

3 1. Motivation: Enumerative geometry Enumerative Geometry: Counting algebro-geometric objects with certain properties. Tropical approach. - Define suitable tropical objects. - Prove that (# of alg. geom. objects) = (# of tropical objects) - Count tropical objects. - Don t shy away from the combinatorics. Actually count them! What we do. We apply this philosophy to the Severi degrees of (some) toric surfaces. We prove they are polynomial and provide some explicit formulas. Federico Ardila (SF State) Severi degrees of toric surfaces December 15, / 23

4 1. Motivation: Counting plane curves. Question How many (possibly reducible) nodal algebraic curves in CP 2 have degree d,. δ nodes, and pass through the right number ( (d+3)d 2 δ) of generic points? Number of such curves is the Severi degree N d,δ. N 1,0 = #{lines through 2 points} = 1. N 2,1 = #{1-nodal conics through 4 points} = 3 N 4,4 = #{4-nodal quartics through 10 points} = 666. If d δ + 2, such a curve is irreducible by Bézout s Theorem, so N d,δ = Gromov-Witten invariant N d,g, with g = (d 1)(d 2) 2 δ. Federico Ardila (SF State) Severi degrees of toric surfaces December 15, / 23

5 1. Motivation: Counting plane curves. Theorem (Fomin Mikhalkin, 2009) For a fixed δ, we have N d,δ = N δ (d), for a combinatorially defined polynomial N δ (d) Q[d], provided d 2δ. Conjectured by Di Francesco Itzykson (1994) and by Göttsche (1997). - J. Steiner (1848): N d,1 = 3(d 1) 2 - A. Cayley (1863): N d,2 = 3 2 (d 1)(d 2)(3d 2 3d 11) - S. Roberts (1867): δ = 3 - I. Vainsencher (1995): δ = 4, 5, 6 - S. Kleiman R. Piene (2001): δ = 7, 8 - F. Block (2010): δ = 9, 10, 11, 12, 13, 14 Federico Ardila (SF State) Severi degrees of toric surfaces December 15, / 23

6 2. What we do: Counting curves on toric surfaces. Goal: Study the analogous problem for toric surfaces. Given a lattice polygon = (c, d), get: (c=slopes, d=lengths) - a (projective) toric surface S = Tor(c). (think: c=surface) - a(n ample) line bundle L = L c (d) on S (think: d= multidegree ) - a (complete) linear system L The Severi degree N,δ is: the degree of the Severi variety {δ-nodal curves in L }. the number of curves in the torus (C ) 2 with δ nodes and Newton polygon, that pass through Z 2 1 δ generic points in (C ) 2 Federico Ardila (SF State) Severi degrees of toric surfaces December 15, / 23

7 2. What we do: Counting curves on toric surfaces. Goal: Study the Severi degrees N,δ for toric surfaces. Ex. Σ m = Hirzebruch surface, b 1 a 1 m + 1 (a, b) = bidegree (non-trivial) slopes: c = m lengths: d = (a, b) N,δ = # curves in (C ) 2 with δ nodes and Newton polygon passing through Z 2 1 δ generic points We find: N,0 = 1, N,1 = 3b 2 m + 6ab 2bm 4a 4b + 4, N,2 = 9 2 b4 m 2 6b 3 m abm a 2 b 2 24abm 2. +2b 2 m 2 12bm 3 24a 2 b Note: Severi degrees are polynomial. In what? In a and b, and also in m! Federico Ardila (SF State) Severi degrees of toric surfaces December 15, / 23

8 2. What we do: Counting curves on toric surfaces. Our main result. Assume (c, d) is h-transverse: (slopes of normal fan are integers) c = ((3, 1, 0, 1); ( 2, 0, 1, 2)) (slopes surface ) d = ((1, 2, 1, 1); (1, 1, 1, 2); 1) (lengths degree ) Theorem (Polynomiality of Severi degrees) (A. - Block, 2010) Fix m, n 1 and δ 1. There exists a universal and combinatorially defined polynomial p δ (c, d) such that the Severi degree N,δ is given by N,δ = p δ (c, d). (1) for any sufficiently large and spread out c Z m+n and d Z m+n+1 0. Federico Ardila (SF State) Severi degrees of toric surfaces December 15, / 23

9 2. Comparing with the Göttsche-Yau-Zaslow formula. There exist universal power series B 1 (q) and B 2 (q) such that the Severi degrees N δ (S, L) of any smooth surface S and sufficiently ample L are δ 0 N δ (S, L)(DG 2 (τ)) δ = (DG 2(τ)/q) χ(l) B 1 (q) K 2 S B 2 (q) LK S ( (τ)d 2 G 2 (τ)/q 2 ) χ(o S )/2 q = e 2πiτ, D = q d, G2(τ) = 1 + ( ) d q 24 n>0 d n d q n (2nd Eisenstein series) (τ) = q k>0 (1 qk ) 24 (Weierstrass -function) O S = structure sheaf, K S =canonical bundle, C 2=2nd Chern class. χ(o S ) = 1 (K 2 12 S + C 2(S)) χ(l) = 1 2 (L2 LK S ) + 1 (K 2 12 S + C 2(S)) Theorem (Göttsche-Yau-Zaslow 1998, Tzeng 2010) If S is smooth and L is sufficiently ample, for a universal polynomial T δ. N δ (S, L) = T δ (L 2, LK S, K 2 S, C 2(S)) Federico Ardila (SF State) Severi degrees of toric surfaces December 15, / 23,

10 2. Comparing with the Göttsche-Yau-Zaslow formula. Theorem (Göttsche-Yau-Zaslow 1998, Tzeng 2010) If S is smooth and L is sufficiently ample, for a universal polynomial T δ. N δ (S, L) = T δ (L 2, LK S, K 2 S, C 2(S)) Think: Severi degrees are topological. In principle, GYZ formula is much more general than ours. In the toric case, our formula is much more general. (S not smooth.) Question Is there a Göttsche-Yau-Zaslow formula for (some) singular surfaces? Federico Ardila (SF State) Severi degrees of toric surfaces December 15, / 23

11 3.1. Proof: Tropical curves Mikhalkin s Correspondence Theorem (2005) replaces enumeration of our algebraic curves: by weighted enumeration of tropical curves dual to subdivisions of : 2 Federico Ardila (SF State) Severi degrees of toric surfaces December 15, / 23

12 3.2. Proof: Floor diagrams Mikhalkin-Brugalle replace enum. of tropical curves with enum. of combinatorial gadgets, (marked) floor diagrams. (If is h-transverse) Federico Ardila (SF State) Severi degrees of toric surfaces December 15, / 23

13 3.2. Proof: Floor diagrams D l, D r : left and right slopes (M of them) t: top length D l = {0, 0, 0, 0}, D r = {1, 1, 1, 0}, t = Marked -floor diagram: permutations l, r of D l, D r weak ordered partition s of t (t = s s M ) a bipartite graph on ({,..., }, {,...,,,..., }) (with M s), linearly ordered and directed L-to-R, with Z + edge weights such that: s are sources/sinks, s have one incoming and one outgoing edge, div( ) = 0, div( ) = 1 or 1, div( i) (r l + s) i. 2 2 r = (1, 1, 0, 1), l = (0, 0, 0, 0), s = (0, 1, 0, 0), r l + s = (1, 2, 0, 1) Federico Ardila (SF State) Severi degrees of toric surfaces December 15, / 23

14 3.2. Proof: Floor diagrams. Theorem (Brugallé Mikhalkin 2007) N,δ = weight(d) δ(d)=δ summing over all marked -floor diagrams D with cogenus δ(d) = δ. 2 2 r = (1, 1, 0, 1), l = (0, 0, 0, 0), s = (0, 1, 0, 0) weight(d) = e edge weight(e) cogenus δ(d) := int( ) Z 2 genus(d) = δ. This is: - literally true for small δ, when curves are irreducible. - essentially true in general, but δ(d) is more complicated. Federico Ardila (SF State) Severi degrees of toric surfaces December 15, / 23

15 3.2. Proof: Floor diagrams. Theorem (Brugallé Mikhalkin 2007) N,δ = weight(d), δ(d)=δ summing over all marked -floor diagrams D with cogenus δ. 2 2 r = (1, 1, 0, 1), l = (0, 0, 0, 0), s = (0, 1, 0, 0) Ok, this is combinatorial, but unmanageable in many ways! Floor diagrams are complicated. Some of the difficulties: - l, r permutations of sets of a variable length - s partition of a variable number - floor diagrams are hard to control - number of vertices isn t even fixed. - allowed edges depend delicately on l, r. Federico Ardila (SF State) Severi degrees of toric surfaces December 15, / 23

16 3.3. Proof: Template decomposition 2 2 r = (1, 1, 0, 1), l = (0, 0, 0, 0), s = (0, 1, 0, 0) Decompose marked floor diagrams into templates: Drop gray vertices, merge white vertices into a top and a bottom one. Drop short edges and isolated black vertices. Keep track of starting points of templates. This process is reversible Federico Ardila (SF State) Severi degrees of toric surfaces December 15, / 23

17 3.3. Proof: Template decomposition N,δ = D : δ(d)=δ weight(d) permutations l, r of D l, D r -floor diagram D templates Γ = (Γ 1,..., Γ m ) starting points k = (k 1,..., k m ). This change of variables effectively decouples (l, r) and Γ. Theorem (Fomin Mikhalkin 2009 (plane curves) / A. Block 2010) N,δ = P Γ (a, k)weight(γ) Γ (l,r) : δ(γ)+δ(l,r)=δ k A(Γ,a) The set of possible Γs is now finite. The set of possible (l, r) is complicated. If c, d are large, l, r are trivial. Federico Ardila (SF State) Severi degrees of toric surfaces December 15, / 23

18 3.3. Proof: Template decomposition (c, d large) N,δ = D : δ(d)=δ weight(d) floor diagram D templates Γ = (Γ 1,..., Γ m ) starting points k = (k 1,..., k m ). Theorem (Fomin Mikhalkin 2009 (plane curves) / A. Block 2010) N,δ = P Γ (a, k)weight(γ) Γ:δ(Γ)=δ k A(Γ,a) # of preimages: combinatorially defined function P Γ (a, k). A(Γ, a) = set of possible k (depends on Γ, c, d). The set of possible Γs is now finite. What about the inner sum? Federico Ardila (SF State) Severi degrees of toric surfaces December 15, / 23

19 It turns out that P Γ (a, k) is a product of binomial coeffs, which is polynomial in a. A(Γ, a) is generally unmanageable. But if d is large enough, it is the set of lattice points in a polytope with fixed facet directions and variable facet parameters. Federico Ardila (SF State) Severi degrees of toric surfaces December 15, / Proof: Discrete integrals over polytopes floor diagram D templates Γ = (Γ 1,..., Γ m ) starting points k = (k 1,..., k m ). P Γ (a, k) = # of preimages of (Γ, k). A(Γ, a) = {possible k for given Γ, a} (a = r l + s) Lemma f Γ (a) = k A(Γ,a) Z m P Γ (a, k) is polynomial.

20 3.4. Proof: Discrete integrals over polytopes Lemma f Γ (a) = k A(Γ,a) Z m P Γ (a, k) is polynomial P Γ (a, k): polynomial in a. A(Γ, a): polytope with fixed facet directions and variable parameters. Then f Γ (a) is the discrete integral of a polynomial over a polytope, so it is piecewise quasipolynomial. piecewise? An (intricate) analysis shows we stay in one region of quasipolynomiality. quasi? Check A(Γ, a) is a facet-unimodular polytope, so we get a polynomial. Federico Ardila (SF State) Severi degrees of toric surfaces December 15, / 23

21 3. The main result, again. Theorem (Polynomiality of Severi degrees) (A.-Block, 2010.) Fix m, n 1 and δ 1 and an h-transverse toric surface Tor(c). There exists a combinatorially defined polynomial p δ,c (d) such that the Severi degree N,δ is given by N,δ = p δ,c (d). for any sufficiently large and spread out d Z m+n+1 0. With a bit more work, Theorem (Universal Polynomiality) (A.-Block, 2010.) Fix m, n 1 and δ 1. There is a universal and combinatorially defined polynomial p δ (c, d) such that the Severi degree N,δ of any h-transverse toric surface is given by N,δ = p δ (c, d). for any sufficiently large and spread out c Z m+n and d Z m+n+1 0. Federico Ardila (SF State) Severi degrees of toric surfaces December 15, / 23

22 4. Comments / questions / future directions. Only for h-transverse polygons? Probably not. We inherit this restriction from Brugalle-Mikhalkin (2007): (curves) (tropical curves) (floor diagrams) (combinatorics) Problem. Modify floor diagrams to make this work in general. Federico Ardila (SF State) Severi degrees of toric surfaces December 15, / 23

23 4. Comments / questions / future directions. Only for h-transverse polygons? Probably not. We inherit this restriction from Brugalle-Mikhalkin (2007): (curves) (tropical curves) (floor diagrams) (combinatorics) Problem. Modify floor diagrams to make this work in general. Only for large c and d? Yes. But the thresholds can probably be improved. Federico Ardila (SF State) Severi degrees of toric surfaces December 15, / 23

24 4. Comments / questions / future directions. Only for h-transverse polygons? Probably not. We inherit this restriction from Brugalle-Mikhalkin (2007): (curves) (tropical curves) (floor diagrams) (combinatorics) Problem. Modify floor diagrams to make this work in general. Only for large c and d? Yes. But the thresholds can probably be improved. Only for toric surfaces? No. Göttsche Yau Zaslow Formula: For S smooth, L sufficiently ample, N δ = T δ (L 2, LK S, K 2 S, c 2(S)), for a universal polynomial T δ. Problem. Find a GYZ formula for S not necessarily smooth. Obvious things don t work: MacPherson s c 2 (S) Resolution of singularities There is surely much more to this story... Federico Ardila (SF State) Severi degrees of toric surfaces December 15, / 23

25 Muchas gracias por su atención. The paper and the slides are available at: Federico Ardila (SF State) Severi degrees of toric surfaces December 15, / 23