E. Shustin, “A tropical approach to enumerative geometry”, Algebra i Analiz, 17:2 (2005), 170–214; St. Petersburg Math. J., 17:2 (2006), 343

Algebra i Analiz

RUS ENG

JOURNALS PEOPLE ORGANISATIONS CONFERENCES SEMINARS VIDEO LIBRARY PACKAGE AMSBIB

JavaScript is disabled in your browser. Please switch it on to enable full functionality of the website

	General information
	Latest issue
	Archive
	Impact factor

	Search papers
	Search references

	RSS
	Latest issue
	Current issues
	Archive issues
	What is RSS

Algebra i Analiz:
Year:
Volume:
Issue:
Page:
	Find

Personal entry:
Login:
Password:
	Save password
	Enter
	Forgotten password?
	Register

Algebra i Analiz, 2005, Volume 17, Issue 2, Pages 170–214 (Mi aa665)

This article is cited in 57 scientific papers (total in 57 papers)

Research Papers

A tropical approach to enumerative geometry

E. Shustin

Tel Aviv University, School of Mathematical Sciences, Aviv, Tel Aviv, Israel

Full-text PDF (2342 kB) Citations (57)

References:

PDF

HTML

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 [18]. 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.

Received: 20.06.2003

English version:
St. Petersburg Mathematical Journal, 2006, Volume 17, Issue 2, Pages 343–375
DOI: https://doi.org/10.1090/S1061-0022-06-00908-3

Bibliographic databases:

Document Type: Article

Language: English

Citation: E. Shustin, “A tropical approach to enumerative geometry”, Algebra i Analiz, 17:2 (2005), 170–214; St. Petersburg Math. J., 17:2 (2006), 343–375

Citation in format AMSBIB

\Bibitem{Shu05}

\by E. Shustin

\paper A~tropical approach to enumerative geometry

\jour Algebra i Analiz

\yr 2005

\vol 17

\issue 2

\pages 170--214

\mathnet{http://mi.mathnet.ru/aa665}

\mathscinet{http://mathscinet.ams.org/mathscinet-getitem?mr=2159589}

\zmath{https://zbmath.org/?q=an:1100.14046}

\transl

\jour St. Petersburg Math. J.

\yr 2006

\vol 17

\issue 2

\pages 343--375

\crossref{https://doi.org/10.1090/S1061-0022-06-00908-3}

Linking options:

https://www.mathnet.ru/eng/aa665

https://www.mathnet.ru/eng/aa/v17/i2/p170

This publication is cited in the following 57 articles:

Eugenii Shustin, “Tropical vertex and real enumerative geometry”, Journal of London Math Soc, 109:1 (2024)
Eugenii Shustin, “Enumeration of non-nodal real plane rational curves”, Math. Z., 307:4 (2024)
Thomas Blomme, “Tropical curves in abelian surfaces II: Enumeration of curves in linear systems”, Trans. Amer. Math. Soc., 376:8 (2023), 5641
Thomas Blomme, “Floor diagrams and enumerative invariants of line bundles over an elliptic curve”, Compositio Math., 159:8 (2023), 1741
Thomas Blomme, “An imaginary refined count for some real rational curves”, Beitr Algebra Geom, 64:3 (2023), 721
Blomme T., “Refined Count For Rational Tropical Curves in Arbitrary Dimension”, Math. Ann., 382:3-4 (2022), 1199–1244
Lang L., Tyomkin I., “A Note on the Severi Problem For Toric Surfaces”, Math. Ann., 2022
Uriel Sinichkin, “Enumeration of algebraic and tropical singular hypersurfaces”, Trans. Amer. Math. Soc., 375:12 (2022), 8529
Hülya Argüz, Pierrick Bousseau, “Real Log Curves in Toric Varieties, Tropical Curves, and Log Welschinger Invariants”, Annales de l'Institut Fourier, 72:4 (2022), 1547
B. Ya. Kazarnovskii, A. G. Khovanskii, A. I. Esterov, “Newton polytopes and tropical geometry”, Russian Math. Surveys, 76:1 (2021), 91–175
Ganor Ya., Shustin E., “Enumeration of Unicuspidal Curves of Any Degree and Genus on Toric Surfaces”, Int. Math. Res. Notices, 2021
St. Petersburg Math. J., 32:2 (2021), 279–306
Takahashi T., “Tropicalization of 1-Tacnodal Curves on Toric Surfaces”, Osaka J. Math., 57:2 (2020), 451–491
Takahashi T., “A Note on Tropical Curves and the Newton Diagrams of Plane Curve Singularities”, Tokyo J. Math., 42:1 (2019), 51–61
Brugalle E., Degtyarev A., Itenberg I., Mangolte F., “Real Algebraic Curves With Large Finite Number of Real Points”, Eur. J. Math., 5:3, SI (2019), 686–711
Mikhalkin G., “Examples of Tropical-to-Lagrangian Correspondence”, Eur. J. Math., 5:3, SI (2019), 1033–1066
Esterov A., “Characteristic Classes of Affine Varieties and Plucker Formulas For Affine Morphisms”, J. Eur. Math. Soc., 20:1 (2018), 15–59
Markwig H., Markwig T., Shustin E., “Enumeration of Complex and Real Surfaces Via Tropical Geometry”, Adv. Geom., 18:1 (2018), 69–100
Itenberg I., Kharlamov V., Shustin E., “Relative Enumerative Invariants of Real Nodal Del Pezzo Surfaces”, Sel. Math.-New Ser., 24:4 (2018), 2927–2990
Leviant P., Shustin E., “Morsifications of Real Plane Curve Singularities”, J. Singul., 18 (2018), 307–328

Citing articles in Google Scholar: Russian citations, English citations
Related articles in Google Scholar: Russian articles, English articles

Statistics & downloads:
Abstract page:	831
Full-text PDF :	444
References:	95
First page:	1

Registration to the website

Logotypes