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Characteristic classes of affine varieties and Plücker formulas for affine morphisms. (English) Zbl 1472.14006

Summary: An enumerative problem on a variety \(V\) is usually solved by reduction to intersection theory in the cohomology of a compactification of \(V\). However, if the problem is invariant under a “nice” group action on \(V\) (so that \(V\) is spherical), then many authors suggested a better home for intersection theory: the direct limit of the cohomology rings of all equivariant compactifications of \(V\). We call this limit the affine cohomology of \(V\) and construct affine characteristic classes of subvarieties of a complex torus, taking values in the affine cohomology of the torus.{ }This allows us to make the first steps in computing affine Thom polynomials. Classical Thom polynomials count how many fibers of a generic proper map of a smooth variety have a prescribed collection of singularities and our affine version addresses the same question for generic polynomial maps of affine algebraic varieites. This notion is also motivated by developing an intersection-theoretic approach to tropical correspondence theorems: they can be reduced to the computation of affine Thom polynomials, because the fundamental class of a variety in the affine cohomology is encoded by the tropical fan of this variety.{ }The first concrete answer that we obtain is the affine version of what were, historically speaking, the first three Thom polynomials–the Plücker formulas for the degree and the number of cusps and nodes of a projectively dual curve. This, in particular, characterizes toric varieties whose projective dual is a hypersurface, computes the tropical fan of the variety of double tangent hyperplanes to a toric variety, and describes the Newton polytope of the hypersurface of non-Morse polynomials of a given degree. We also make a conjecture on the general form of affine Thom polynomials; a key ingredient is the \(n\)-ary fan, generalizing the secondary polytope.

MSC:

14C17 Intersection theory, characteristic classes, intersection multiplicities in algebraic geometry
32M10 Homogeneous complex manifolds
14T15 Combinatorial aspects of tropical varieties
14M25 Toric varieties, Newton polyhedra, Okounkov bodies
14M10 Complete intersections
14N15 Classical problems, Schubert calculus
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[1] Aluffi, P.: Characteristic classes of discriminants and enumerative geometry. Comm. Algebra 26, 3165-3193 (1998)Zbl 0934.14039 MR 1641591 · Zbl 0934.14039
[2] Aluffi, P.: Characteristic classes of singular varieties. In: Topics in Cohomological Studies of Algebraic Varieties, Trends in Math., Birkh¨auser, 1-32 (2005)MR 2143071 · Zbl 1487.14019
[3] Aluffi, P.: Chern classes of graph hypersurfaces and deletion-contraction. Moscow Math. J. 12, 671-700 (2012)Zbl 1291.14014 MR 3076849 · Zbl 1291.14014
[4] Antipova, I., Tsikh, A.: The discriminant locus of a system of n Laurent polynomials in n variables. Izv. Math. 76, 881-906 (2012)Zbl 1254.32012 MR 3024862 · Zbl 1254.32012
[5] Arnold, V. I., Gusein-Zade, S. M., Varchenko, A. N.: Singularities of Differentiable Maps. Vol. I, Birkh¨auser (2012)Zbl 1290.58001 MR 2896292 · Zbl 1290.58001
[6] Bernstein, D. N.: The number of roots of a system of equations. Funct. Anal. Appl. 9, 183-185 (1975)Zbl 0328.32001 MR 0435072 · Zbl 0328.32001
[7] Bertrand, B., Brugalle, E., Mikhalkin, G.: Genus 0 characteristic numbers of the tropical projective plane. Compos. Math. 150, 46-104 (2014)Zbl 06333812 MR 3164359 · Zbl 1375.14209
[8] Billera, L. J., Sturmfels, B.: Fiber polytopes. Ann. of Math. (2) 135, 527-549 (1992) Zbl 0762.52003 MR 1166643 · Zbl 0762.52003
[9] Brion, M.: Piecewise polynomial functions, convex polytopes and enumerative geometry. In: Parameter Spaces (Warszawa, 1994), Banach Center Publ. 36, Inst. Math., Polish Acad. Sci., Warszawa, 25-44 (1996)Zbl 0878.14035 MR 1481477 · Zbl 0878.14035
[10] Brion, M.: The structure of the polytope algebra. Tohoku Math. J. (2) 49, 1-32 (1997) Zbl 0881.52008 MR 1431267 Characteristic classes of affine varieties57 · Zbl 0881.52008
[11] Brion, M.: Groupe de Picard et nombres caract´eristiques des vari´et´es sph´eriques. Duke Math. J. 58, 397-424 (1989)Zbl 0701.14052 MR 1016427 · Zbl 0701.14052
[12] Brion, M., Joshua, R.: Equivariant Chow ring and Chern classes of wonderful symmetric varieties of minimal rank. Transformation Groups 13, 471-493 (2008) Zbl 1172.14031 MR 2452601 · Zbl 1172.14031
[13] Brion, M., Kausz, I.: Vanishing of top equivariant Chern classes of regular embeddings. Asian J. Math. 9, 489-496 (2005)Zbl 1119.14044 MR 2216242 · Zbl 1119.14044
[14] Casagrande, C., Di Rocco, S.: Projective Q-factorial toric varieties covered by lines. Comm. Contemp. Math. 10, 363-389 (2008)Zbl 1165.14036 MR 2417921 · Zbl 1165.14036
[15] Catanese, F.: On Severi’s proof of the double point formula. Comm. Algebra 7, 763- 773 (1979)Zbl 0411.14016 MR 0529319 [CC+13]Cattani, E., Cueto, M. A., Dickenstein, A., Di Rocco, S., Sturmfels, B.: Mixed discriminants. Math. Z. 274, 761-778 (2013)Zbl 1273.13051 MR 3078246
[16] Curran, R., Cattani, E.: Restriction of A-discriminants and dual defect varieties. J. Symbolic Comput. 42, 115-135 (2007)Zbl 1167.14326 MR 2284288 · Zbl 1167.14326
[17] Danilov, V. I.: The geometry of toric varieties. Russian Math. Surveys 33, 97-154 (1978)Zbl 0425.14013 MR 0495499 · Zbl 0425.14013
[18] De Concini, C., Procesi, C.: Complete symmetric varieties. II. Intersection theory. In: Algebraic Groups and Related Topics, Adv. Stud. Pure Math. 6, North-Holland, 481- 513 (1985)Zbl 0596.14041 MR 0803344 · Zbl 0596.14041
[19] De Loera, J. A., Liu, F., Yoshida, R.: A generating function for all semi-magic squares and the volume of the Birkhoff polytope. J. Algebraic Combin. 30, 113-139 (2009) Zbl 1187.05009 MR 2519852 · Zbl 1187.05009
[20] Dickenstein, A., Di Rocco, S., Piene, R.: Higher order duality and toric embeddings. Ann. Inst. Fourier (Grenoble) 64, 375-400 (2014)Zbl 1329.14098 MR 3330552 · Zbl 1329.14098
[21] Dickenstein, A., Emiris, I., Karasoulou, A.: Plane mixed discriminants and toric Jacobians. In: SAGA—Advances in ShApes, Geometry and Algebra, Geom. Comput. 10, Springer, 105-121 (2014)Zbl 1331.65047 MR 3289657 · Zbl 1331.65047
[22] Dickenstein, A., Feichtner, E. M., Sturmfels, B.: Tropical discriminants. J. Amer. Math. Soc. 20, 1111-1133 (2007)Zbl 1166.14033 MR 2328718 · Zbl 1166.14033
[23] Di Rocco, S.: Projective duality of toric manifolds and defect polytopes. Proc. London Math. Soc. 3, 85-104 (2006)Zbl 1098.14039 MR 2235483 · Zbl 1098.14039
[24] Esterov, A.: On the existence of mixed fiber bodies. Moscow Math. J. 8, 433-442 (2008)Zbl 1160.52004 MR 2483219 · Zbl 1160.52004
[25] Esterov, A.: Newton polyhedra of discriminants of projections. Discrete Comput. Geom. 44, 96-148 (2010)Zbl 1273.14105 MR 2639821 · Zbl 1273.14105
[26] Esterov, A. I.: Tropical varieties with polynomial weights and corner loci of piecewise polynomials. Moscow Math. J. 12, 55-76 (2012)Zbl 1271.14092 MR 2952426 · Zbl 1271.14092
[27] Esterov, A.: Multiplicities of degenerations of matrices and mixed volumes of Cayley polyhedra. J. Singularities 6, 27-36 (2012)Zbl 1292.52007 MR 2971304 · Zbl 1292.52007
[28] Esterov, A.: The discriminants of a system of equations. Adv. Math. 245, 534-572 (2013)Zbl 1291.13044 MR 3084437 · Zbl 1291.13044
[29] Esterov, A., Khovanskii, A. G.: Elimination theory and Newton polytopes. Funct. Anal. Other Math. 2, 45-71 (2008)Zbl 1192.14038 MR 2466086 · Zbl 1192.14038
[30] Fulton, W., MacPherson, R., Sottile, F., Sturmfels, B.: Intersection theory on spherical varieties. J. Algebraic Geom. 4, 181-194 (1995)Zbl 0819.14019 MR 1299008 · Zbl 0819.14019
[31] Fulton, W., Sturmfels, B.: Intersection theory on toric varieties. Topology 36, 335-353 (1997)Zbl 0885.14025 MR 1415592 58Alexander Esterov · Zbl 0885.14025
[32] Gathmann, A., Kerber, M., Markwig, H.: Tropical fans and the moduli spaces of tropical curves. Compos. Math. 145, 173-195 (2009)Zbl 1169.51021 MR 2480499 · Zbl 1169.51021
[33] Gelfand, I. M., Kapranov, M. M., Zelevinsky, A. V.: Discriminants, Resultants, and Miltidimensional Determinants. Birkh¨auser (1994)Zbl 0827.14036 MR 1264417 · Zbl 0827.14036
[34] Gonz´alez P´erez, P. D.: Singularit´es quasi-ordinaires toriques et poly‘edre de Newton du discriminant. Canad. J. Math. 52, 348-368 (2000)Zbl 0970.14027 MR 1755782 · Zbl 0970.14027
[35] G¨ottsche, L.: A conjectural generating function for numbers of curves on surfaces. Comm. Math. Phys. 196, 523-533 (1998)Zbl 0934.14038 MR 1645204 · Zbl 0934.14038
[36] Greuel, G.-M., Lossen, C., Shustin, E.: Singular Algebraic Curves. Springer (2015) · Zbl 1411.14001
[37] Gross, A.: Correspondence theorems via tropicalizations of moduli spaces. Comm. Contemp. Math. 18, 1550043 (2016)Zbl 06568929 MR 3477404 · Zbl 1387.14152
[38] Gusein-Zade, S. M., Luengo, I., Melle-Hern´andez, A.: Partial resolutions and the zetafunction of a singularity. Comment. Math. Helv. 72, 244-256 (1997)Zbl 0901.32024 MR 1470090 · Zbl 0901.32024
[39] Huh, J., Sturmfels, B.: Likelihood geometry. In: Combinatorial Algebraic Geometry, Lecture Notes in Math. 2108, Springer, 63-117 (2014).Zbl 1328.14004 MR 3329087 · Zbl 1328.14004
[40] Katz, E.: Tropical invariants from the secondary fan. Adv. Geom. 9, 153-180 (2009) Zbl 1184.14093 MR 2523837 · Zbl 1184.14093
[41] Katz, E., Payne, S.: Piecewise polynomials, Minkowski weights, and localization on toric varieties. Algebra Number Theory 2, 135-155 (2008)Zbl 1158.14042 MR 2377366 · Zbl 1158.14042
[42] Katz, E., Payne, S.: Realization spaces for tropical fans. In: Combinatorial Aspects of Commutative Algebra and Algebraic Geometry, Abel Sympos. 6, Springer, 73-88 (2011)Zbl 1248.14066 MR 2810427 · Zbl 1248.14066
[43] Kazarian, M.: Multisingularities, cobordisms, and enumerative geometry. Russian Math. Surveys 58, 665-724 (2003) (the journal’s translation contains multiple mistakes; see the author’s translation athttp://www.mi.ras.ru/˜kazarian/papers/multie.ps) · Zbl 1062.58039
[44] Kazarnovski˘ı, B.: Truncation of systems of polynomial equations, ideals and varieties. Izv. Math. 63, 535-547 (1999)MR 1712124 · Zbl 1041.17028
[45] Kazarnovski˘ı, B.: c-fans and Newton polyhedra of algebraic varieties. Izv. Math. 67, 439-460 (2003)Zbl 1077.14072 MR 1992192 · Zbl 1077.14072
[46] Khovanskii, A. G.: Newton polyhedra and toroidal varieties. Funct. Anal. Appl. 11, 289-296 (1977)Zbl 0445.14019 MR 0476733 · Zbl 0445.14019
[47] Khovanskii, A. G.: Newton polyhedra and the genus of complete intersections. Funct. Anal. Appl. 12, 38-46 (1978)Zbl 0406.14035 MR 0487230 · Zbl 0406.14035
[48] Pukhlikov, A. V., Khovanskii, A. G.: Finitely additive measures of virtual polyhedra. St. Petersburg Math. J. 4, 337-356 (1993)MR 1182399 · Zbl 0791.52010
[49] Kiritchenko, V.: Chern classes for reductive groups and an adjunction formula. Ann. Inst. Fourier (Grenoble) 56, 1225-1256 (2006)Zbl 1120.14005 MR 2266889 · Zbl 1120.14005
[50] Kouchnirenko, A. G.: Poly‘edres de Newton et nombres de Milnor. Invent. Math. 32, 1-32 (1976)Zbl 0328.32007 MR 0419433 · Zbl 0328.32007
[51] Lando, S., Zvonkin, K.: Graphs on Surfaces and Their Applications. Springer (2004) Zbl 1040.05001 MR 2036721 · Zbl 1040.05001
[52] MacPherson, R. D.: Chern classes for singular algebraic varieties. Ann. of Math. (2) 100, 423-432 (1974)Zbl 0311.14001 MR 0361141 · Zbl 0311.14001
[53] McMullen, P.: Weights on polytopes. Discrete Comput. Geom. 15, 363-388 (1996) Zbl 0849.52011 MR 1384882 Characteristic classes of affine varieties59 · Zbl 0849.52011
[54] McMullen, P.: Mixed fibre polytopes. Discrete Comput. Geom. 32, 521-532 (2004) Zbl 1087.52006 MR 2096746 · Zbl 1087.52006
[55] McMullen, P.: The polytope algebra. Adv. Math. 78, 76-130 (1989)Zbl 0686.52005 MR 1021549 · Zbl 0686.52005
[56] Mikhalkin, G.: Enumerative tropical algebraic geometry in R2. J. Amer. Math. Soc. 18, 313-377 (2005)Zbl 1092.14068 MR 2137980 · Zbl 1092.14068
[57] Mikhalkin, G.: Tropical geometry and its applications. In: Proc. ICM (Madrid, 2006), Eur. Math. Soc., 827-852 (2006)Zbl 1103.14034 MR 2275625 · Zbl 1103.14034
[58] Morelli, R.: A theory of polyhedra. Adv. Math. 97, 1-73 (1993)Zbl 0779.52016 MR 1200289 · Zbl 0779.52016
[59] Orevkov, S.: The volume of the Newton polytope of a discriminant. Russ. Math. Surveys 54, 1033-1034 (1999)Zbl 0962.14032 MR 1741673 · Zbl 0962.14032
[60] Schwartz, M.-H.: Classes et caract‘eres de Chern-Mather des espaces lin´eaires. C. R. Acad. Sci. Paris S´er. I Math. 295, 399-402 (1982)Zbl 0513.14007 MR 0684735 · Zbl 0513.14007
[61] Severi, F.: Sulle intersezioni delle variet‘a algebriche e sopra i loro caratteri e singolarit‘a proiettive. Mem. Accad. Sci. Torino Cl. Sci. Fis. Mat. Natur. 52, 61-118 (1902) JFM 34.0699.01 · JFM 34.0699.01
[62] Shitov, Y.: When do the r-by-r minors of a matrix form a tropical basis? J. Combin. Theory Ser. A 120, 1166-1201 (2013)Zbl 1300.14065 MR 3044537 · Zbl 1300.14065
[63] Shustin, E.: A tropical approach to enumerative geometry. St. Petersburg Math. J. 17, 343-375 (2006)Zbl 1100.14046 MR 2159589 · Zbl 1100.14046
[64] Shustin, E.: Tropical and algebraic curves with multiple points. In: Perspectives in Analysis, Geometry, and Topology, Progr. Math. 296, Birkh¨auser/Springer, 431-464 (2012)Zbl 1291.14084 MR 2884046 · Zbl 1291.14084
[65] Siebert, B., Nishinou, T.: Toric degenerations of toric varieties and tropical curves. Duke Math. J. 135, 1-51 (2006)Zbl 1105.14073 MR 2259922 · Zbl 1105.14073
[66] Stanley, R.: Generalized H-vectors, intersection cohomology of toric varieties, and related results. In: Commutative Algebra and Combinatorics (Kyoto, 1985), Adv. Stud. Pure Math. 11, North-Holland, 187-213 (1987)Zbl 0652.52007 MR 0951205 · Zbl 0652.52007
[67] Sturmfels, B.: On the Newton polytope of the resultant. J. Algebraic Combin. 3, 207- 236 (1994)Zbl 0798.05074 MR 1268576 · Zbl 0798.05074
[68] Sturmfels, B., Tevelev, J.: Elimination theory for tropical varieties. Math. Res. Lett. 15, 543-562 (2008)Zbl 1157.14038 MR 2407231 · Zbl 1157.14038
[69] Sturmfels, B., Tevelev, E. A., Yu, J.: The Newton polytope of the implicit equation. Moscow Math. J. 7, 327-346 (2007)MR 2387885 Zbl 1133.13026 · Zbl 1133.13026
[70] Tevelev, E.: Compactifications of subvarieties of tori. Amer. J. Math. 129, 1087-1104 (2007)Zbl 1154.14039 MR 2343384 · Zbl 1154.14039
[71] Timorin, V. A.: On polytopes that are simple at the edges. Funct. Anal. Appl. 35, 189- 198 (2001)Zbl 1009.52020 MR 1864987 · Zbl 1009.52020
[72] Tyomkin, I.: Tropical geometry and correspondence theorems via toric stacks. Math. Ann. 353, 945-995 (2012)Zbl 1272.14045 MR 2923954 · Zbl 1272.14045
[73] Tzeng, Y.-J.: A proof of the G¨ottsche-Yau-Zaslow formula. J. Differential Geom. 90, 439-472 (2012)Zbl 1253.14054 MR 2916043 · Zbl 1253.14054
[74] Viro, O.: Some integral calculus based on Euler characteristic. In: Topology and Geometry—Rohlin Seminar, Lecture Notes in Math. 1346, Springer, 127-138 (1988) Zbl 0686.14019 MR 0970076 · Zbl 0686.14019
[75] Yang, J. J.: Tropical Severi varieties. Portugal. Math. 70, 59-91 (2013) Zbl 1312.14147 MR 3074395 · Zbl 1312.14147
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