St. Petersburg Mathematical Journal

			Journals Home Search Author Info Subscribe Tech Support My Subscriptions Help
ISSN 1547-7371(e) ISSN 1061-0022(p)

Previous issue \| Table of contents \| Next issue Recently posted articles \| Most recent issue \| All issues Previous Article

A tropical approach to enumerative geometry

Author(s): E. Shustin
Original publication: Algebra i Analiz, tom 17 (2005), vypusk 2.
Journal: St. Petersburg Math. J. 17 (2006), 343-375.
MSC (2000): Primary 14H15; Secondary 12J25, 14H20, 14M25, 14N10
Posted: February 20, 2006
MathSciNet review: 2159589
Retrieve article in: PDF

Abstract | References | Similar articles | Additional information

Abstract: A detailed algebraic-geometric background is presented for the tropical approach to enumeration of singular curves on toric surfaces, which consists of reducing the enumeration of algebraic curves to that of non-Archimedean amoebas, the images of algebraic curves by a real-valued non-Archimedean valuation. This idea was proposed by Kontsevich and recently realized by Mikhalkin, who enumerated the nodal curves on toric surfaces. The main technical tools are a refined tropicalization of one-parametric equisingular families of curves and the patchworking construction of singular algebraic curves. The case of curves with a cusp and the case of real nodal curves are also treated.

References:

1.: L. Caporaso and J. Harris, Counting plane curves of any genus, Invent. Math. 131 (1998), no. 2, 345-392. MR 1608583 (99i:14064)
2.: -, Parameter spaces for curves on surfaces and enumeration of rational curves, Compositio Math. 113 (1998), no. 2, 155-208. MR 1639183 (99e:14061a)
3.: Xi Chen, Rational curves on K3 surfaces, J. Algebraic Geom. 8 (1999), 245-278. MR 1675158 (2000d:14057)
4.: L. Chiantini and C. Ciliberto, On the Severi varieties on surfaces in $\bold P\sp3$ , J. Algebraic Geom. 8 (1999), no. 1, 67-83. MR 1658208 (2000f:14082)
5.: S. Diaz and J. Harris, Ideals associated to deformations of singular plane curves, Trans. Amer. Math. Soc. 309 (1988), no. 2, 433-468. MR 0961600 (89m:14003)
6.: M. Forsberg, M. Passare, and A. Tsikh, Laurent determinants and arrangements of hyperplane amoebas, Adv. Math. 151 (2000), 45-70. MR 1752241 (2001m:32060)
7.: I. M. Gel'fand, M. M. Kapranov, and A. V. Zelevinsky, Discriminants, resultants, and multidimensional determinants, Birkhäuser Boston, Inc., Boston, MA, 1994. MR 1264417 (95e:14045)
8.: A. Henriques, An analogue of convexity for complements of amoebas of varieties of higher codimensions, Preprint, Berkeley, May 2001.
9.: A. Hirschowitz, La méthode d'Horace pour l'interpolation à plusieurs variables, Manuscripta Math. 50 (1985), 337-388. MR 0784148 (86j:14013)
10.: I. Itenberg and E. Shustin, Singular points and limit cycles of planar polynomial vector fields, Duke Math. J. 102 (2000), no. 1, 1-37. MR 1741776 (2001d:34019)
11.: -, Viro theorem and topology of real and complex combinatorial hypersurfaces, Israel J. Math. 133 (2003), 189-238. MR 1968429 (2004c:14115)
12.: M. M. Kapranov, Amoebas over non-Archimedean fields, Preprint, 2000.
13.: M. Kontsevich and Y. Soibelman, Homological mirror symmetry and torus fibrations, Preprint arXiv:math.SG/0011041.
14.: M. Kontsevich and Yu. Tschinkel, Non-Archimedean Kähler geometry (in preparation).
15.: G. Mikhalkin, Real algebraic curves, the moment map and amoebas, Ann. of Math. (2) 151 (2000), no. 1, 309-326. MR 1745011 (2001c:14083)
16.: -, Amoebas of algebraic varieties, Preprint arXiv:math.AG/0108225
17.: G. Mikhalkin and H. Rullgård, Amoebas of maximal area, Internat. Math. Res. Notices 2001, no. 9, 441-451. MR 1829380 (2002b:14079)
18.: G. Mikhalkin, Counting curves via lattice paths in polygons, C. R. Math. Acad. Sci. Paris 336 (2003), no. 8, 629-634. MR 1988122 (2004d:14077)
19.: -, Enumerative tropical algebraic geometry in $\mathbb{R}^2$ , Preprint arXiv: math AG/0312530.
20.: M. Passare and H. Rullgård, Amoebas, Monge-Ampère measures and triangulations of the Newton polytope, Res. Rep. in Math. no. 10, Stockholm Univ., 2000.
21.: H. Rullgård, Stratification des espaces de polynômes de Laurent et la structure de leurs amibes, C. R. Acad. Sci. Paris Sér. I Math. 331 (2000), 355-358. MR 1784913 (2001h:32044)
22.: E. Shustin, Real plane algebraic curves with prescribed singularities, Topology 32 (1993), 845-856. MR 1241875 (95f:14049)
23.: -, Surjectivity of the resultant map: A solution to the inverse Bezout problem, Comm. Algebra 23 (1995), no. 3, 1145-1163. MR 1316754 (96a:12004)
24.: -, Critical points of real polynomials, subdivisions of Newton polyhedra and topology of real algebraic hypersurfaces, Topology of Real Algebraic Varieties and Related Topics, Amer. Math. Soc. Transl. (2), vol. 173, Amer. Math. Soc., Providence, RI, 1996, pp. 203-223. MR 1384319 (97g:14040)
25.: -, Gluing of singular and critical points, Topology 37 (1998), no. 1, 195-217. MR 1480886 (99f:14073)
26.: -, Lower deformations of isolated hypersurface singularities, Algebra i Analiz 11 (1999), no. 5, 221-249; English transl., St. Petersburg Math. J. 11 (2000), no. 5, 883-908. MR 1734355 (2000m:32039)
27.: -, Patchworking singular algebraic curves, non-Archimedean amoebas and enumerative geometry, Preprint arXiv: math.AG/0211278.
28.: E. Shustin and I. Tyomkin, Patchworking singular algebraic curves, to appear in Israel J. Math.
29.: B. Sturmfels, Solving systems of polynomial equations, CBMS Regional Conf. Ser. in Math., vol. 97, Amer. Math. Soc., Providence, RI, 2002. MR 1925796 (2003i:13037)
30.: O. Ya. Viro, Gluing of algebraic hypersurfaces, smoothing of singularities and construction of curves, Proceedings of the Leningrad International Topological Conference (Aug. 1982), ``Nauka'', Leningrad, 1983, pp. 149-197. (Russian)
31.: -, Gluing of plane real algebraic curves and constructions of curves of degrees 6 and 7, Topology (Leningrad, 1982), Lecture Notes in Math., vol. 1060, Springer, Berlin, 1984, pp. 187-200. MR 0770238 (87i:14029)
32.: -, Real algebraic plane curves: Constructions with controlled topology, Algebra i Analiz 1 (1989), no. 5, 1-73; English transl., St. Petersburg Math. J. 1 (1990), no. 5, 1059-1134. MR 1036837 (91b:14078)
33.: -, Patchworking real algebraic varieties, Preprint at http://www.math.uu.se/ $\sim$ oleg/preprints. html (1995).
34.: J. Wahl, Equisingular deformations of plane algebraic curves, Trans. Amer. Math. Soc. 193 (1974), 143-170. MR 0419439 (54:7460)
35.: J.-Y. Welschinger, Invariants of real rational symplectic 4-manifolds and lower bounds in real enumerative geometry, C. R. Math. Acad. Sci. Paris 336 (2003), 341-344. MR 1976315 (2004m:53157)

	Publications Meetings The Profession Membership Programs Math Samplings Policy & Advocacy In the News About the AMS
\|