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On abelian varieties the theta divisor of which is singular in codimension 3. (Sur les variétés abéliennes dont le diviseur thêta est singulier en codimension 3.) (French) Zbl 0699.14058
This paper investigates the loci \({\mathcal N}_ k^ g\) in the moduli space \({\mathcal A}_ g\) of principally polarized \(g\)-dimensional abelian varieties \((A,\Theta)\). It consists of the points in \({\mathcal A}_ g\) corresponding to \((A,\Theta)\)’s for which the principal polarization \(\Theta\) satisfies \(\dim(\text{Sing}(\Theta))\geq k\).
Important applications of these loci can e.g. be found in Mumford’s work, who used the divisor \({\mathcal N}^ g_ 0\) consisting of those \((A,\Theta)\) with \(\Theta\) singular to prove that \({\mathcal A}_ g\) is of general type for \(g\geq 7\). Also Andreotti and Mayer used such loci in connection with the Schottky problem; they proved that the locus \(\bar {\mathcal I}_ g\) which is the closure in \({\mathcal A}_ g\) of the set of Jacobians of smooth curves of genus \(g\) is an irreducible component of \({\mathcal N}^ g_{g-4}\) for \(g\geq 4\).

However, it is known that this does not completely characterize Jacobians. Other irreducible components of \({\mathcal N}^ g_{g-4}\) and also irreducible components of some other \({\mathcal N}_ k^ g\)’s are given. For \(g=5\) this allows the author to give a complete description of all irreducible components of \({\mathcal N}^ 5_ 1.\)
In the paper two constructions for describing such components are given: Firstly, one can take special Prym varieties \(P(\tilde C/C)\), which are g-dimensional abelian varieties associated with unramified double covers \(\tilde C\to C\) of curves C of genus \(g-1\). A description for which covers one obtains \(P(\tilde C/C)\) with \(\text{codim(Sing}(\Theta))\leq 4\) is due to Beauville. The necessary information about their image in \({\mathcal A}_ g\) is provided by Donagi.
Secondly, the author uses certain pairs \((A,\Theta)\) in which \(A\) is isogenous to a product of abelian varieties. It turns out that all except two of the components found in terms of Prym varieties are in fact subsets of irreducible components which can be described by such non-simple abelian varieties.

MSC:
14K25 Theta functions and abelian varieties
14K30 Picard schemes, higher Jacobians
14K10 Algebraic moduli of abelian varieties, classification
14H40 Jacobians, Prym varieties
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[1] E. Arbarello, M. Cornalba, P. A. Griffiths, and J. Harris, Geometry of Algebraic Curves, I , Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 267, Springer-Verlag, New York, 1985. · Zbl 0559.14017
[2] A. Andreotti and A. L. Mayer, On period relations for abelian integrals on algebraic curves , Ann. Scuola Norm. Sup. Pisa (3) 21 (1967), 189-238. · Zbl 0222.14024 · numdam:ASNSP_1967_3_21_2_189_0 · eudml:83420
[3] W. Barth, Fortsetzung, meromorpher Funktionen in Tori und komplex-projektiven Räumen , Invent. Math. 5 (1968), 42-62. · Zbl 0159.37602 · doi:10.1007/BF01404537 · eudml:141904
[4] A. Beauville, Prym varieties and the Schottky problem , Invent. Math. 41 (1977), no. 2, 149-196. · Zbl 0333.14013 · doi:10.1007/BF01418373 · eudml:142490
[5] A. Beauville, Variétés de Prym et jacobiennes intermédiaires , Ann. Sci. École Norm. Sup. (4) 10 (1977), no. 3, 309-391. · Zbl 0368.14018 · numdam:ASENS_1977_4_10_3_309_0 · eudml:81998
[6] A. Beauville, Sous-variétés spéciales des variétés de Prym , Compositio Math. 45 (1982), no. 3, 357-383. · Zbl 0504.14022 · numdam:CM_1982__45_3_357_0 · eudml:89541
[7] A. Beauville and O. Debarre, Une relation entre deux approches du problème de Schottky , Invent. Math. 86 (1986), no. 1, 195-207. · Zbl 0659.14021 · doi:10.1007/BF01391500 · eudml:143392
[8] P. Deligne and D. Mumford, The irreducibility of the space of curves of given genus , Inst. Hautes Études Sci. Publ. Math. (1969), no. 36, 75-109. · Zbl 0181.48803 · doi:10.1007/BF02684599 · numdam:PMIHES_1969__36__75_0 · eudml:103899
[9] R. Donagi, The tetragonal construction , Bull. Amer. Math. Soc. (N.S.) 4 (1981), no. 2, 181-185. · Zbl 0491.14016 · doi:10.1090/S0273-0979-1981-14875-8
[10] R. Donagi and R. C. Smith, The structure of the Prym map , Acta Math. 146 (1981), no. 1-2, 25-102. · Zbl 0538.14019 · doi:10.1007/BF02392458
[11] A. Grothendieck, Éléments de géométrie algébrique. II. Étude globale élémentaire de quelques classes de morphismes , Inst. Hautes Études Sci. Publ. Math. (1961), no. 8, 222. · Zbl 0118.36206 · doi:10.1007/BF02684778 · numdam:PMIHES_1960__4__5_0 · numdam:PMIHES_1961__8__5_0 · numdam:PMIHES_1961__11__5_0
[12] W. Fulton and R. Lazarsfeld, On the connectedness of degeneracy loci and special divisors , Acta Math. 146 (1981), no. 3-4, 271-283. · Zbl 0469.14018 · doi:10.1007/BF02392466
[13] R. Friedman and R. Smith, Degenerations of Prym varieties and intersections of three quadrics , Invent. Math. 85 (1986), no. 3, 615-635. · Zbl 0619.14027 · doi:10.1007/BF01390330 · eudml:143382
[14] R. Friedman and R. Smith, The generic Torelli theorem for the Prym map , Invent. Math. 67 (1982), no. 3, 473-490. · Zbl 0506.14042 · doi:10.1007/BF01398932 · eudml:142922
[15] M. Green, Quadrics of rank four in the ideal of a canonical curve , Invent. Math. 75 (1984), no. 1, 85-104. · Zbl 0542.14018 · doi:10.1007/BF01403092 · eudml:143090
[16] J. Harris, Theta-characteristics on algebraic curves , Trans. Amer. Math. Soc. 271 (1982), no. 2, 611-638. · Zbl 0513.14025 · doi:10.2307/1998901
[17] J. I. Igusa, A desingularization problem in the theory of Siegel modular functions , Math. Ann. 168 (1967), 228-260. · Zbl 0145.09702 · doi:10.1007/BF01361555 · eudml:161503
[18] J. I. Igusa, Theta Functions , Grundlehren der Math. Wiss., vol. 194, Springer-Verlag, Berlin-New York, 1972. · Zbl 0251.14016
[19] L. Masiewicki, Universal properties of Prym varieties with an application to algebraic curves of genus five , Trans. Amer. Math. Soc. 222 (1976), 221-240. JSTOR: · Zbl 0333.14012 · doi:10.2307/1997667 · links.jstor.org
[20] D. Mumford, On the Kodaira Dimension of the Siegel Modular Variety , Algebraic geometry-open problems (Ravello, 1982), Springer Lecture Notes, vol. 997, Springer-Verlag, New York, 1983, pp. 348-375. · Zbl 0527.14036 · doi:10.1007/BFb0061652
[21] D. Mumford, Prym Varieties I , Contributions to Analysis (a collection of papers dedicated to Lipman Bers), Academic Press, New York, 1974, pp. 325-350. · Zbl 0299.14018
[22] D. Mumford, Theta characteristics of an algebraic curve , Ann. Sci. École Norm. Sup. (4) 4 (1971), 181-192. · Zbl 0216.05904 · numdam:ASENS_1971_4_4_2_181_0 · eudml:81878
[23] D. Mumford, Varieties defined by quadratic equations , Questions on Algebraic Varieties (C.I.M.E., III Ciclo, Varenna, 1969), Edizioni Cremonese, Rome, 1970, pp. 29-100. · Zbl 0198.25801
[24] D. Mumford, On the equations defining abelian varieties I , Invent. Math. 1 (1966), 287-354. · Zbl 0219.14024 · doi:10.1007/BF01389737 · eudml:141872
[25] D. Mumford, Abelian Varieties , Tata Studies in Math., vol. 5, Published for the Tata Institute of Fundamental Research, Bombay; Oxford University Press, London, London, 1970. · Zbl 0223.14022
[26] D. Mumford, An analytic construction of degenerating abelian varieties over complete rings , Compositio Math. 24 (1972), 239-272. · Zbl 0241.14020 · numdam:CM_1972__24_3_239_0 · eudml:89120
[27] 1 Y. Namikawa, A new compactification of the Siegel space and degeneration of Abelian varieties. I , Math. Ann. 221 (1976), no. 2, 97-141. · Zbl 0306.14016 · doi:10.1007/BF01433145 · eudml:162837
[28] 2 Y. Namikawa, A new compactification of the Siegel space and degeneration of Abelian varieties. II , Math. Ann. 221 (1976), no. 3, 201-241. · Zbl 0327.14013 · doi:10.1007/BF01596390 · eudml:162844
[29] S. Recillas, Jacobians of curves with \(g^1_4\)’s are the Prym’s of trigonal curves , Bol. Soc. Mat. Mexicana (2) 19 (1974), no. 1, 9-13. · Zbl 0343.14012
[30] R. Smith and R. Varley, Components of the locus of singular theta divisors of genus \(5\) , Algebraic Geometry, Sitges (Barcelona) 1983, Lecture Notes in Math., vol. 1124, Springer-Verlag, Berlin-New York, 1985, pp. 338-416. · Zbl 0598.14036 · doi:10.1007/BFb0075005
[31] M. Teixidor i Bigas, For which Jacobi varieties is \(\mathrm Sing\,\Theta\) reducible? J. Reine Angew. Math. 354 (1984), 141-149. · Zbl 0542.14020 · doi:10.1515/crll.1984.354.141 · crelle:GDZPPN002201976 · eudml:152680
[32] A. N. Tjurin, Five lectures on three-dimensional varieties , Uspehi Mat. Nauk 27 (1972), no. 5, (167), 3-50. · Zbl 0263.14012 · doi:10.1070/rm1972v027n05ABEH001384
[33] A. N. Tjurin, The intersection of quadrics , Uspehi Mat. Nauk 30 (1975), no. 6(186), 51-99. · Zbl 0339.14020 · doi:10.1070/RM1975v030n06ABEH001530
[34] A. N. Tjurin, The geometry of the Poincaré theta divisor of a Prym variety , Math. USSR-Izv 9 (1975), 951-986. · Zbl 0339.14017 · doi:10.1070/IM1975v009n05ABEH001513
[35] G. Welters, A theorem of Gieseker-Petri type for Prym varieties , Ann. Sci. École Norm. Sup. (4) 18 (1985), no. 4, 671-683. · Zbl 0628.14036 · numdam:ASENS_1985_4_18_4_671_0 · eudml:82169
[36] G. Welters, The surface \(C-C\) on Jacobi varieties and 2nd order theta functions , Acta Math. 157 (1986), no. 1-2, 1-22. · Zbl 0771.14012 · doi:10.1007/BF02392589
[37] G. Welters, Abel-Jacobi isogenies for certain types of Fano threefolds , Mathematical Centre Tracts, vol. 141, Mathematisch Centrum, Amsterdam, 1981. · Zbl 0474.14028
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