Abstract
We prove Welters’ trisecant conjecture: an indecomposable principally polarized abelian variety $X$ is the Jacobian of a curve if and only if there exists a trisecant of its Kummer variety $K(X)$.
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@article {arb-decon, MRKEY = {750718},
AUTHOR = {Arbarello, Enrico and De Concini, Corrado},
TITLE = {On a set of equations characterizing {R}iemann matrices},
JOURNAL = {Ann. of Math.},
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AUTHOR = {Arbarello, Enrico and Krichever, Igor and Marini, Giambattista},
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} -
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author = {Krichever, I. M.},
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VOLUME={36},
YEAR={1981},
PAGES={72--77},
} -
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ISSN = {0002-3264},
MRCLASS = {58F07 (14K25 35Q20 39A70)},
MRNUMBER = {87c:58049},
MRREVIEWER = {Alexander A. Pankov},
ZBLNUMBER = {0603.39004},
} -
[kr-schot]
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@incollection {kr-schot, MRKEY = {2263198},
AUTHOR = {Krichever, I. M.},
TITLE = {Integrable linear equations and the {R}iemann-{S}chottky problem},
BOOKTITLE = {Algebraic Geometry and Number Theory},
SERIES = {Progr. Math.},
NUMBER = {253},
PAGES = {497--514},
PUBLISHER = {Birkhäuser},
ADDRESS = {Boston, MA},
YEAR = {2006},
MRCLASS = {14H70 (14H42 37K20)},
MRNUMBER = {2007h:14049},
MRREVIEWER = {Armando K. Treibich},
DOI = {10.1007/978-0-8176-4532-8_8},
ZBLNUMBER = {1132.14032},
} -
[prym] I. M. Krichever, "A characterization of Prym varieties," Int. Math. Res. Not., vol. 2006, p. I, 2006.
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[bete]
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[n-kr]
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AUTHOR = {Krichever, I. M. and Phong, D. H.},
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[zab]
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JOURNAL = {Uspekhi Mat. Nauk},
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VOLUME = {50},
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NUMBER = {6(306)},
PAGES = {3--56},
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} -
[mar]
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AUTHOR = {Mumford, D.},
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VENUE={{K}yoto, 1977},
PAGES = {115--153},
PUBLISHER = {Kinokuniya Book Store},
ADDRESS = {Tokyo},
YEAR = {1978},
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[shiota]
T. Shiota, "Characterization of Jacobian varieties in terms of soliton equations," Invent. Math., vol. 83, iss. 2, pp. 333-382, 1986.
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AUTHOR = {Shiota, Takahiro},
TITLE = {Characterization of {J}acobian varieties in terms of soliton equations},
JOURNAL = {Invent. Math.},
FJOURNAL = {Inventiones Mathematicae},
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@article {wel, MRKEY = {731833},
AUTHOR = {Welters, G. E.},
TITLE = {On flexes of the {K}ummer variety (note on a theorem of {R}. {C}. {G}unning)},
JOURNAL = {Nederl. Akad. Wetensch. Indag. Math.},
FJOURNAL = {Koninklijke Nederlandse Akademie van Wetenschappen. Indagationes Mathematicae},
VOLUME = {45},
YEAR = {1983},
NUMBER = {4},
PAGES = {501--520},
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} -
[wel1]
G. E. Welters, "A criterion for Jacobi varieties," Ann. of Math., vol. 120, iss. 3, pp. 497-504, 1984.
@article {wel1, MRKEY = {769160},
AUTHOR = {Welters, G. E.},
TITLE = {A criterion for {J}acobi varieties},
JOURNAL = {Ann. of Math.},
FJOURNAL = {Annals of Mathematics. Second Series},
VOLUME = {120},
YEAR = {1984},
NUMBER = {3},
PAGES = {497--504},
ISSN = {0003-486X},
CODEN = {ANMAAH},
MRCLASS = {14H40 (14K10)},
MRNUMBER = {86e:14011},
MRREVIEWER = {David Ortland},
DOI = {10.2307/1971084},
ZBLNUMBER = {0574.14027},
}