Computing Node Polynomials for Plane Curves Florian Block - PowerPoint PPT Presentation

Computing Node Polynomials for Plane Curves Florian Block University of Michigan FPSAC 2010 August 5, 2010 http://www-personal.umich.edu/ ∼ blockf/NodePolyTalk.pdf arXiv:1006.0218 Florian Block (U of Michigan) Computing Node Polynomials August 5, 2010 1 / 14

Combinatorial Rules in Enumerative Geometry Enumerative Geometry: counting geometric objects with certain properties. Strategy: reduce the problem to enumeration of combinatorial gadgets. Example: the Littlewood - Richardson - Rule. Why are such rules needed? This talk: Enumeration of plane curves via (marked) floor diagrams.         � �     ✲ # = # . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ✉ ❡ ✉ ✉ ❡ ❡ . . . . . . . . . ✉ ❡ ❡ ✉ ✉ ❡ . . . . . . . . ✉ ❡ ❡ ✉ . . . . . . . . . . . . . . . . . . . . . . . . . . ✲ ✲ ✲ ✲ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .   . . . . . . . . . . ✲ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .         Florian Block (U of Michigan) Computing Node Polynomials August 5, 2010 2 / 14

Counting plane curves Question How many (possibly reducible) nodal algebraic curves in CP 2 have degree d, δ nodes, and pass through ( d +3) d − δ generic points? 2 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, a curve is irreducible by B´ ezout’s Theorem, so N d ,δ = Gromov-Witten invariant N d , g , with g = ( d − 1)( d − 2) − δ . 2 Florian Block (U of Michigan) Computing Node Polynomials August 5, 2010 3 / 14

Node Polynomials 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 ≥ 2 δ . Polynomiality was conjectured by P. Di Francesco–C. Itzykson (1994) and by L. G¨ ottsche (1997). The N δ ( d ) are called node polynomials. We have deg( N δ ( d )) = 2 δ . Florian Block (U of Michigan) Computing Node Polynomials August 5, 2010 4 / 14

Chronology of Node Polynomials N d , 1 = 3( d − 1) 2 J. Steiner (1848): ( d ≥ 1) N d , 2 = 3 2 ( d − 1)( d − 2)(3 d 2 − 3 d − 11) A. Cayley (1863): ( d ≥ 1) S. Roberts (1867): N d , 3 = 9 2 d 6 − 27 d 5 + 9 2 d 4 + 423 2 d 3 − 229 d 2 − 829 2 d + 525 ( d ≥ 3) I. Vainsencher (1995): δ = 4 , 5 , 6 S. Kleiman–R. Piene (2001): δ = 7 , 8 Florian Block (U of Michigan) Computing Node Polynomials August 5, 2010 5 / 14

New Node Polynomials Theorem (B.) The node polynomials N δ ( d ) , for δ = 9 , 10 , 11 , 12 , 13 , 14 , are given by . . . . Florian Block (U of Michigan) Computing Node Polynomials August 5, 2010 6 / 14

New Node Polynomials Theorem (B.) The node polynomials N δ ( d ) , for δ = 9 , 10 , 11 , 12 , 13 , 14 , are given by . . . . 19683 19683 6561 1751787 4529277 562059 d 26 + d 28 − d 27 − d 25 − d 24 − d 23 N 14 ( d ) = 358758400 12812800 2562560 3942400 1971200 9856 398785599 5214288411 4860008991 63174295089 332872084467 d 22 + d 19 + d 21 − d 20 − d 18 + 788480 1254400 89600 358400 89600 3103879378581 4913807521304691 899178800016807 279086438050359453 d 16 + d 15 + d 17 − d 14 + 985600 27596800 8968960 44844800 468967272863997483 318443311640108577 328351365725506869 d 12 + d 13 − d 11 − 51251200 1971200 985600 1120586814080571923 9448861028448843949 30880785216736406143 d 10 − d 9 − d 8 + 358400 1254400 689920 444525313669622586903 11429038221675466251 269709254062572016617 d 7 + d 6 − d 5 + 3942400 24640 246400 74660630664748878665353 140531359469510983018159 16863931195154225977601 d 4 + d 3 + d 2 − 22422400 22422400 1121120 64314454486825349085 d − 32644422296329680 . − 4004 Based on an algorithm of S. Fomin–G. Mikhalkin, with improvements. Maple calculation of N 14 ( d ) took 70 days. Florian Block (U of Michigan) Computing Node Polynomials August 5, 2010 6 / 14

Polynomiality Threshold Define d ∗ ( δ ) = smallest d ∗ such that d ≥ d ∗ ( δ ) implies N d ,δ = N δ ( d ). S. Fomin–G. Mikhalkin: d ∗ ( δ ) ≤ 2 δ for δ ≥ 1. Theorem (B.) For δ ≥ 1 , we have d ∗ ( δ ) ≤ δ . ottsche (1997) conjectured: d ∗ ( δ ) ≤ ⌈ δ L. G¨ 2 ⌉ + 1. Theorem (B.) For 3 ≤ δ ≤ 14 , we have d ∗ ( δ ) = ⌈ δ 2 ⌉ + 1 . For δ ≤ 8, this was established by S. Kleiman–R. Piene (2001). Florian Block (U of Michigan) Computing Node Polynomials August 5, 2010 7 / 14

Leading terms of the node polynomials Theorem (B.) For any δ , the nine leading terms of N δ ( d ) are: 3 δ δ ( δ − 4) δ ( δ − 1)(20 δ − 13) » d 2 δ − 2 + d 2 δ − 2 δ d 2 δ − 1 − d 2 δ − 3 + N δ ( d ) = δ ! 3 6 δ ( δ − 1)(69 δ 2 − 85 δ + 92) δ ( δ − 1)( δ − 2)(702 δ 2 − 629 δ − 286) d 2 δ − 4 − d 2 δ − 5 + − 54 270 δ ( δ − 1)( δ − 2)(6028 δ 3 − 15476 δ 2 + 11701 δ + 4425) d 2 δ − 6 + + 3240 δ ( δ − 1)( δ − 2)( δ − 3)(13628 δ 3 − 6089 δ 2 − 29572 δ − 24485) d 2 δ − 7 + + 11340 δ ( δ − 1)( δ − 2)( δ − 3)(282855 δ 4 − 931146 δ 3 + 417490 δ 2 + 425202 δ + 1141616) # d 2 δ − 8 + · · · . − 204120 The first 7 terms were conjectured by P. Di Francesco–C. Itzykson (1994). Florian Block (U of Michigan) Computing Node Polynomials August 5, 2010 8 / 14

Main Technique: (Marked) Floor Diagrams Mikhalkin’s Correspondence Theorem (2005) replaces enumeration of algebraic plane curves by weighted enumeration of tropical plane curves. E. Brugall´ e–S. Fomin–G. Mikhalkin (2007, 2009) reduced the latter to enumeration of combinatorial gadgets called (marked) floor diagrams.   ✉ ✲ ✲ ✲ ✲           ❡             ✉ . . . . .     . . . . . . . . . . . . . .     . . . . . . . . . .     . . . . . . . . . . . . . . . . # = # = # { ❡ . . . } . . . . . . . . . . . . . . . . . . . ✲ . . . . . . . . . . . . . . . . . . . . . . ✉ . . .     . . . . . . . . . . . . . . . . . . . . . .     . . . . . . . . . . . . . . . . . . .     . . . . . . . . . . . . . . . . . . . .     . . . . . . . . . . . . . . . . . . . . . . . . .     ✲ . . . . ❡ . . . . . . . . . . . . . . . . . . .   . . . . . . . . . . . . . . . . . . .   . . . . . . . . . . . . . . . . . . . . .   . . . . . . . . . . . . . . . . . . . . . .   . . . . . . ❡ . . . .   . . . . . . . . . .   . . . . . . . . . . . . . . . . . . . . . . . . . . . ❡ Florian Block (U of Michigan) Computing Node Polynomials August 5, 2010 9 / 14