Characterizing Jacobians via trisecants of the Kummer variety (2022)

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 $K(X)$.

  • [arb-decon] Characterizing Jacobians via trisecants of the Kummer variety (1) E. Arbarello and C. De Concini, "On a set of equations characterizing Riemann matrices," Ann. of Math., vol. 120, iss. 1, pp. 119-140, 1984.

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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.},
    FJOURNAL = {Annals of Mathematics. Second Series},
    VOLUME = {120},
    YEAR = {1984},
    NUMBER = {1},
    PAGES = {119--140},
    ISSN = {0003-486X},
    CODEN = {ANMAAH},
    MRCLASS = {14H40 (14K20 14K25 32G20)},
    MRNUMBER = {86a:14025},
    MRREVIEWER = {Gerald E. Welters},
    DOI = {10.2307/2007073},
    ZBLNUMBER = {0551.14016},
    }

  • [arbarello] Characterizing Jacobians via trisecants of the Kummer variety (2) E. Arbarello and C. De Concini, "Another proof of a conjecture of S. P. Novikov on periods of abelian integrals on Riemann surfaces," Duke Math. J., vol. 54, iss. 1, pp. 163-178, 1987.

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    @article {arbarello, MRKEY = {885782},
    AUTHOR = {Arbarello, Enrico and De Concini, Corrado},
    TITLE = {Another proof of a conjecture of {S}. {P}. {N}ovikov on periods of abelian integrals on {R}iemann surfaces},
    JOURNAL = {Duke Math. J.},
    FJOURNAL = {Duke Mathematical Journal},
    VOLUME = {54},
    YEAR = {1987},
    NUMBER = {1},
    PAGES = {163--178},
    ISSN = {0012-7094},
    CODEN = {DUMJAO},
    MRCLASS = {14H40 (32G20 58F07)},
    MRNUMBER = {88i:14025},
    MRREVIEWER = {Bert van Geemen},
    DOI = {10.1215/S0012-7094-87-05412-3},
    ZBLNUMBER = {0629.14022},
    }

  • [flex] E. Arbarello, I. Krichever, and G. Marini, "Characterizing Jacobians via flexes of the Kummer variety," Math. Res. Lett., vol. 13, iss. 1, pp. 109-123, 2006.

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    @article {flex, MRKEY = {2200050},
    AUTHOR = {Arbarello, Enrico and Krichever, Igor and Marini, Giambattista},
    TITLE = {Characterizing {J}acobians via flexes of the {K}ummer variety},
    JOURNAL = {Math. Res. Lett.},
    FJOURNAL = {Mathematical Research Letters},
    VOLUME = {13},
    YEAR = {2006},
    NUMBER = {1},
    PAGES = {109--123},
    ISSN = {1073-2780},
    MRCLASS = {14H42 (14H40)},
    MRNUMBER = {2007b:14064},
    MRREVIEWER = {H. Lange},
    ZBLNUMBER = {1098.14020},
    }

  • [ch1] J. L. Burchnall and T. W. Chaundy, "Commutative ordinary differential operators. I," Proc. London Math. Soc., vol. 21, pp. 420-440, 1922.

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    @article{ch1,
    author={Burchnall, J. L. and Chaundy, T. W.},
    TITLE={Commutative ordinary differential operators. {\rm I}},
    JOURNAL={Proc. London Math. Soc.},
    VOLUME={21},
    YEAR={1922},
    PAGES={420--440},
    }

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    @article{ch2,
    author={Burchnall, J. L. and Chaundy, T. W.},
    TITLE={Commutative ordinary differential operators. {\rm II}},
    JOURNAL={Proc. Royal Soc. London},
    VOLUME={118},
    YEAR={1928},
    PAGES={557--583},
    }

  • [mdl] Characterizing Jacobians via trisecants of the Kummer variety (3) P. Deligne and D. Mumford, "The irreducibility of the space of curves of given genus," Inst. Hautes Études Sci. Publ. Math., iss. 36, pp. 75-109, 1969.

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    @article {mdl, MRKEY = {0262240},
    AUTHOR = {Deligne, P. and Mumford, D.},
    TITLE = {The irreducibility of the space of curves of given genus},
    JOURNAL = {Inst. Hautes Études Sci. Publ. Math.},
    FJOURNAL = {Institut des Hautes Études Scientifiques. Publications Mathématiques},
    NUMBER = {36},
    YEAR = {1969},
    PAGES = {75--109},
    ISSN = {0073-8301},
    MRCLASS = {14.20},
    MRNUMBER = {41 \#6850},
    MRREVIEWER = {Manfred Herrmann},
    URL = {http://www.numdam.org/item?id=PMIHES_1969__36__75_0},
    ZBLNUMBER = {0181.48803},
    }

  • [fay] J. D. Fay, Theta Functions on Riemann Surfaces, New York: Springer-Verlag, 1973, vol. 352.

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    @book {fay, MRKEY = {0335789},
    AUTHOR = {Fay, John D.},
    TITLE = {Theta Functions on {R}iemann Surfaces},
    SERIES = {Lecture Notes in Math.},
    VOLUME={352},
    PUBLISHER = {Springer-Verlag},
    ADDRESS = {New York},
    YEAR = {1973},
    PAGES = {iv+137},
    MRCLASS = {30A48 (14H15)},
    MRNUMBER = {49 \#569},
    MRREVIEWER = {H. M. Farkas},
    ZBLNUMBER = {0281.30013},
    }

  • [gun1] Characterizing Jacobians via trisecants of the Kummer variety (4) R. C. Gunning, "Some curves in abelian varieties," Invent. Math., vol. 66, iss. 3, pp. 377-389, 1982.

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    @article {gun1, MRKEY = {662597},
    AUTHOR = {Gunning, R. C.},
    TITLE = {Some curves in abelian varieties},
    JOURNAL = {Invent. Math.},
    FJOURNAL = {Inventiones Mathematicae},
    VOLUME = {66},
    YEAR = {1982},
    NUMBER = {3},
    PAGES = {377--389},
    ISSN = {0020-9910},
    CODEN = {INVMBH},
    MRCLASS = {14K10 (14H15 14K25 32G20)},
    MRNUMBER = {83i:14033},
    MRREVIEWER = {H. H. Martens},
    DOI = {10.1007/BF01389218},
    ZBLNUMBER = {0485.14009},
    }

  • [kr1] I. M. Krichever, "Integration of nonlinear equations by methods of algebraic geometry," Funct. Anal. Appl., vol. 11, pp. 12-26, 1977.

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    @article{kr1,
    author = {Krichever, I. M.},
    TITLE={Integration of nonlinear equations by methods of algebraic geometry},
    JOURNAL={Funct. Anal. Appl.},
    VOLUME={11},
    YEAR={1977},
    PAGES={12--26},
    ZBLNUMBER={0368.35002},
    }

  • [kr2] I. M. Krichever, "Methods of algebraic geometry in the theory of nonlinear equations," Russian Math. Surveys, vol. 32, pp. 185-213, 1977.

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    @article{kr2,
    author = {Krichever, I. M.},
    TITLE={Methods of algebraic geometry in the theory of nonlinear equations},
    JOURNAL={Russian Math. Surveys},
    VOLUME={32},
    YEAR={1977},
    PAGES={185--213},
    ZBLNUMBER={0386.35022},
    }

  • [kr-dif] I. M. Krichever, "Algebraic curves and nonlinear difference equations," Uspekhi Mat. Nauk, vol. 33, iss. 4(202), pp. 215-216, 1978.

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    @article {kr-dif, MRKEY = {510681},
    AUTHOR = {Krichever, I. M.},
    TITLE = {Algebraic curves and nonlinear difference equations},
    JOURNAL = {Uspekhi Mat. Nauk},
    FJOURNAL = {Akademiya Nauk SSSR i Moskovskoe Matematicheskoe Obshchestvo. Uspekhi Matematicheskikh Nauk},
    VOLUME = {33},
    YEAR = {1978},
    NUMBER = {4(202)},
    PAGES = {215--216},
    ISSN = {0042-1316},
    MRCLASS = {58F07 (39A70)},
    MRNUMBER = {80k:58055},
    MRREVIEWER = {Alexander A. Pankov},
    ZBLNUMBER={0382.39003},
    }

  • [kr3] I. M. Krichever, "Elliptic solutions of the Petviashvili equation, and integrable systems of particles," Funktsional. Anal. i Prilozhen., vol. 14, iss. 4, pp. 45-54, 95, 1980.

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    @article {kr3, MRKEY = {595728},
    AUTHOR = {Krichever, I. M.},
    TITLE = {Elliptic solutions of the Petviashvili equation, and integrable systems of particles},
    JOURNAL = {Funktsional. Anal. i Prilozhen.},
    FJOURNAL = {Akademiya Nauk SSSR. Funktsional$'$nyĭAnaliz i ego Prilozheniya},
    VOLUME = {14},
    YEAR = {1980},
    NUMBER = {4},
    PAGES = {45--54, 95},
    ISSN = {0374-1990},
    MRCLASS = {58F07 (14H45 35Q20 70M99)},
    MRNUMBER = {82e:58046},
    MRREVIEWER = {Alexander A. Pankov},
    ZBLNUMBER={0462.35080},
    }

  • [kr-toda] I. M. Krichever, "The periodic nonabelian Toda lattice and two-dimensional generalization, appendix to: B. Dubrovin, Theta functions and non-linear equations," Uspekhi Math. Nauk, vol. 36, pp. 72-77, 1981.

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    @article{kr-toda,
    author = {Krichever, I. M.},
    TITLE={The periodic nonabelian Toda lattice and two-dimensional generalization, appendix to: B. Dubrovin, Theta functions and non-linear equations},
    JOURNAL={Uspekhi Math. Nauk},
    VOLUME={36},
    YEAR={1981},
    PAGES={72--77},
    }

  • [krdd] I. M. Krichever, "Two-dimensional periodic difference operators and algebraic geometry," Dokl. Akad. Nauk SSSR, vol. 285, iss. 1, pp. 31-36, 1985.

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    @article {krdd, MRKEY = {812140},
    AUTHOR = {Krichever, I. M.},
    TITLE = {Two-dimensional periodic difference operators and algebraic geometry},
    JOURNAL = {Dokl. Akad. Nauk SSSR},
    FJOURNAL = {Doklady Akademii Nauk SSSR},
    VOLUME = {285},
    YEAR = {1985},
    NUMBER = {1},
    PAGES = {31--36},
    ISSN = {0002-3264},
    MRCLASS = {58F07 (14K25 35Q20 39A70)},
    MRNUMBER = {87c:58049},
    MRREVIEWER = {Alexander A. Pankov},
    ZBLNUMBER = {0603.39004},
    }

  • [kr-schot] Characterizing Jacobians via trisecants of the Kummer variety (5) I. M. Krichever, "Integrable linear equations and the Riemann-Schottky problem," in Algebraic Geometry and Number Theory, Boston, MA: Birkhäuser, 2006, pp. 497-514.

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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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    @article {prym, MRKEY = {2233714},
    AUTHOR = {Krichever, I. M.},
    TITLE = {A characterization of {P}rym varieties},
    JOURNAL = {Int. Math. Res. Not.},
    FJOURNAL = {International Mathematics Research Notices},
    YEAR = {2006},
    PAGES = {Art. ID 81476, 36},
    ISSN = {1073-7928},
    MRCLASS = {14H40 (14H42 14H70 37K20)},
    MRNUMBER = {2007k:14059},
    MRREVIEWER = {Emma Previato},
    VOLUME = {2006},
    }

  • [bete] Characterizing Jacobians via trisecants of the Kummer variety (6) I. M. Krichever, O. Lipan, P. Wiegmann, and A. Zabrodin, "Quantum integrable models and discrete classical Hirota equations," Comm. Math. Phys., vol. 188, iss. 2, pp. 267-304, 1997.

    (Video) Igor Krichever: Algebraic-geometrical methods in the theory of integrable systems...

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    @article {bete, MRKEY = {1471815},
    AUTHOR = {Krichever, I. M. and Lipan, O. and Wiegmann, P. and Zabrodin, A.},
    TITLE = {Quantum integrable models and discrete classical {H}irota equations},
    JOURNAL = {Comm. Math. Phys.},
    FJOURNAL = {Communications in Mathematical Physics},
    VOLUME = {188},
    YEAR = {1997},
    NUMBER = {2},
    PAGES = {267--304},
    ISSN = {0010-3616},
    CODEN = {CMPHAY},
    MRCLASS = {58F07 (39A10 82B23)},
    MRNUMBER = {99c:58076},
    MRREVIEWER = {Mayumi Ohmiya},
    DOI = {10.1007/s002200050165},
    ZBLNUMBER = {0896.58035},
    }

  • [n-kr] Characterizing Jacobians via trisecants of the Kummer variety (7) I. M. Krichever and S. P. Novikov, "A two-dimensionalized Toda chain, commuting difference operators, and holomorphic vector bundles," Uspekhi Mat. Nauk, vol. 58, iss. 3(351), pp. 51-88, 2003.

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    @article {n-kr, MRKEY = {1998774},
    AUTHOR = {Krichever, I. M. and Novikov, S. P.},
    TITLE = {A two-dimensionalized {T}oda chain, commuting difference operators, and holomorphic vector bundles},
    JOURNAL = {Uspekhi Mat. Nauk},
    FJOURNAL = {Rossiĭskaya Akademiya Nauk. Moskovskoe Matematicheskoe Obshchestvo. Uspekhi Matematicheskikh Nauk},
    VOLUME = {58},
    YEAR = {2003},
    NUMBER = {3(351)},
    PAGES = {51--88},
    ISSN = {0042-1316},
    MRCLASS = {37K60 (14H60 37K20)},
    MRNUMBER = {2004j:37144},
    MRREVIEWER = {Igor Yu. Potemine},
    DOI = {10.1070/RM2003v058n03ABEH000628},
    }

  • [kp] I. M. Krichever and D. H. Phong, "Symplectic forms in the theory of solitons," in Surveys in Differential Geometry: Integral Systems, Somerville: Internat. Press, 1998, vol. IV, pp. 239-313.

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    @incollection {kp, MRKEY = {1726930},
    AUTHOR = {Krichever, I. M. and Phong, D. H.},
    TITLE = {Symplectic forms in the theory of solitons},
    BOOKTITLE = {Surveys in Differential Geometry: Integral Systems},
    SERIES = {Surv. Differ. Geom.},
    VOLUME={IV},
    EDITORS={Terng, C. L. and Uhlenbeck, C. L.},
    PAGES = {239--313},
    PUBLISHER = {Internat. Press},
    ADDRESS={Somerville},
    YEAR = {1998},
    MRCLASS = {37K10 (14H70 35Q51 37K40 53D30)},
    MRNUMBER = {2001k:37114},
    MRREVIEWER = {Ian A. B. Strachan},
    ZBLNUMBER = {0931.35148},
    }

  • [zab] Characterizing Jacobians via trisecants of the Kummer variety (8) I. M. Krichever and A. Zabrodin, "Spin generalization of the Ruijsenaars-Schneider model, the nonabelian two-dimensionalized Toda lattice, and representations of the Sklyanin algebra," Uspekhi Mat. Nauk, vol. 50, iss. 6(306), pp. 3-56, 1995.

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    @article {zab, MRKEY = {1379076},
    AUTHOR = {Krichever, I. M. and Zabrodin, A.},
    TITLE = {Spin generalization of the {R}uijsenaars-{S}chneider model, the nonabelian two-dimensionalized {T}oda lattice, and representations of the {S}klyanin algebra},
    JOURNAL = {Uspekhi Mat. Nauk},
    FJOURNAL = {Rossiĭskaya Akademiya Nauk. Moskovskoe Matematicheskoe Obshchestvo. Uspekhi Matematicheskikh Nauk},
    VOLUME = {50},
    YEAR = {1995},
    NUMBER = {6(306)},
    PAGES = {3--56},
    ISSN = {0042-1316},
    MRCLASS = {58F07 (16S99 81R10 82B23)},
    MRNUMBER = {97f:58068},
    MRREVIEWER = {Alexander V. Shapovalov},
    DOI = {10.1070/RM1995v050n06ABEH002632},
    }

  • [mar] Characterizing Jacobians via trisecants of the Kummer variety (9) G. Marini, "A geometrical proof of Shiota’s theorem on a conjecture of S. P. Novikov," Compositio Math., vol. 111, iss. 3, pp. 305-322, 1998.

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    @article {mar, MRKEY = {1617132},
    AUTHOR = {Marini, Giambattista},
    TITLE = {A geometrical proof of {S}hiota's theorem on a conjecture of {S}. {P}. {N}ovikov},
    JOURNAL = {Compositio Math.},
    FJOURNAL = {Compositio Mathematica},
    VOLUME = {111},
    YEAR = {1998},
    NUMBER = {3},
    PAGES = {305--322},
    ISSN = {0010-437X},
    CODEN = {CMPMAF},
    MRCLASS = {14H42 (14H40 14K25)},
    MRNUMBER = {99a:14039},
    MRREVIEWER = {Emma Previato},
    DOI = {10.1023/A:1000310019510},
    ZBLNUMBER = {0945.14017},
    }

  • [mum] D. Mumford, "An algebro-geometric construction of commuting operators and of solutions to the Toda lattice equation, Korteweg-de Vries equation and related nonlinear equation," in Proc. Internat. Sympos. Algebraic Geometry, Tokyo, 1978, pp. 115-153.

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    @inproceedings {mum, MRKEY = {578857},
    AUTHOR = {Mumford, D.},
    TITLE = {An algebro-geometric construction of commuting operators and of solutions to the {T}oda lattice equation, {K}orteweg-de {V}ries equation and related nonlinear equation},
    BOOKTITLE = {Proc. {I}nternat. {S}ympos. {A}lgebraic {G}eometry},
    VENUE={{K}yoto, 1977},
    PAGES = {115--153},
    PUBLISHER = {Kinokuniya Book Store},
    ADDRESS = {Tokyo},
    YEAR = {1978},
    MRCLASS = {14K25 (14H15 14K10 34C35 34C40 58F07)},
    MRNUMBER = {83j:14041},
    MRREVIEWER = {D. J. Lewis},
    }

  • [wilson] Characterizing Jacobians via trisecants of the Kummer variety (10) G. Segal and G. Wilson, "Loop groups and equations of KdV type," Inst. Hautes Études Sci. Publ. Math., iss. 61, pp. 5-65, 1985.

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    @article {wilson, MRKEY = {783348},
    AUTHOR = {Segal, Graeme and Wilson, George},
    TITLE = {Loop groups and equations of {K}d{V} type},
    JOURNAL = {Inst. Hautes Études Sci. Publ. Math.},
    FJOURNAL = {Institut des Hautes Études Scientifiques. Publications Mathématiques},
    NUMBER = {61},
    YEAR = {1985},
    PAGES = {5--65},
    ISSN = {0073-8301},
    CODEN = {PMIHA6},
    MRCLASS = {58F07 (14K99 35Q20 58G35)},
    MRNUMBER = {87b:58039},
    MRREVIEWER = {A. M. Vinogradov},
    URL = {http://www.numdam.org/item?id=PMIHES_1985__61__5_0},
    ZBLNUMBER = {0592.35112},
    }

  • [serr] Characterizing Jacobians via trisecants of the Kummer variety (11) J. Serre, "Faisceaux algébriques cohérents," Ann. of Math., vol. 61, pp. 197-278, 1955.

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    @article {serr, MRKEY = {0068874},
    AUTHOR = {Serre, Jean-Pierre},
    TITLE = {Faisceaux algébriques cohérents},
    JOURNAL = {Ann. of Math.},
    FJOURNAL = {Annals of Mathematics. Second Series},
    VOLUME = {61},
    YEAR = {1955},
    PAGES = {197--278},
    ISSN = {0003-486X},
    MRCLASS = {14.0X},
    MRNUMBER = {16,953c},
    MRREVIEWER = {C. Chevalley},
    DOI = {10.2307/1969915},
    ZBLNUMBER = {0067.16201},
    }

  • [shiota] Characterizing Jacobians via trisecants of the Kummer variety (12) 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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    @article {shiota, MRKEY = {818357},
    AUTHOR = {Shiota, Takahiro},
    TITLE = {Characterization of {J}acobian varieties in terms of soliton equations},
    JOURNAL = {Invent. Math.},
    FJOURNAL = {Inventiones Mathematicae},
    VOLUME = {83},
    YEAR = {1986},
    NUMBER = {2},
    PAGES = {333--382},
    ISSN = {0020-9910},
    CODEN = {INVMBH},
    MRCLASS = {14H40 (58F07)},
    MRNUMBER = {87j:14047},
    MRREVIEWER = {Bert van Geemen},
    DOI = {10.1007/BF01388967},
    ZBLNUMBER = {0621.35097},
    }

  • [wel] G. E. Welters, "On flexes of the Kummer variety (note on a theorem of R. C. Gunning)," Nederl. Akad. Wetensch. Indag. Math., vol. 45, iss. 4, pp. 501-520, 1983.

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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},
    ISSN = {0019-3577},
    CODEN = {IMTHBJ},
    MRCLASS = {14H40 (14K10)},
    MRNUMBER = {86e:14010},
    MRREVIEWER = {David Ortland},
    }

  • [wel1] Characterizing Jacobians via trisecants of the Kummer variety (13) G. E. Welters, "A criterion for Jacobi varieties," Ann. of Math., vol. 120, iss. 3, pp. 497-504, 1984.

    Show bibtex

    @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},
    }

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