# 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 $K(X)$.

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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}, }

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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}, }

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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}, }

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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}, }

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(Video) Robert Auffarth--The Schottky Problem and secants of the Kummer variety

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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}, }

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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}, }

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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}, }

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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}, }

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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}, }

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(Video) Rationality for Threefolds Over Non-Closed Fields

@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}, }

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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}, }

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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}, }

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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}, }

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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}, }

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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}, }

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(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}, }

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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}, }

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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}, }

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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}, }

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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}, }

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(Video) Bartosz Naskręcki: Elliptic and hyperelliptic realisations of low degree hypergeometric motives

@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}, }

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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}, }

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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}, }

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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}, }

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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}, }

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@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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