On parametrizing exceptional tangent cones to Prym theta divisors
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- by Roy Smith and Robert Varley PDF
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Abstract:
The theta divisor of a Jacobian variety is parametrized by a smooth divisor variety via the Abel map, with smooth projective linear fibers. Hence the tangent cone to a Jacobian theta divisor at any singularity is parametrized by an irreducible projective linear family of linear spaces normal to the corresponding fiber. The divisor variety $X$ parametrizing a Prym theta divisor $\Xi$, on the other hand, is singular over any exceptional point. Hence although the fibers of the Abel Prym map are still smooth, the normal cone in $X$ parametrizing the tangent cone of $\Xi$ can have nonlinear fibers.
In this paper we highlight the diverse and interesting structure that these parametrizing maps can have for exceptional tangent cones to Prym theta divisors. We also propose an organizing framework for the various possible cases. As an illustrative example, we compute the case of a Prym variety isomorphic to the intermediate Jacobian of a cubic threefold, where the projectivized tangent cone, the threefold itself, is parametrized by the 2 parameter family of cubic surfaces cut by hyperplanes through a fixed line on the threefold.
References
- A. Andreotti and A. L. Mayer, On period relations for abelian integrals on algebraic curves, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (3) 21 (1967), 189–238. MR 220740
- E. Arbarello, M. Cornalba, P. A. Griffiths, and J. Harris, Geometry of algebraic curves. Vol. I, Grundlehren der mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 267, Springer-Verlag, New York, 1985. MR 770932, DOI 10.1007/978-1-4757-5323-3
- Arnaud Beauville, Prym varieties and the Schottky problem, Invent. Math. 41 (1977), no. 2, 149–196. MR 572974, DOI 10.1007/BF01418373
- Arnaud Beauville, Variétés de Prym et jacobiennes intermédiaires, Ann. Sci. École Norm. Sup. (4) 10 (1977), no. 3, 309–391 (French). MR 472843, DOI 10.24033/asens.1329
- Arnaud Beauville, Les singularités du diviseur $\Theta$ de la jacobienne intermédiaire de l’hypersurface cubique dans $\textbf {P}^{4}$, Algebraic threefolds (Varenna, 1981) Lecture Notes in Math., vol. 947, Springer, Berlin-New York, 1982, pp. 190–208 (French). MR 672617
- Armand Borel, Linear algebraic groups, W. A. Benjamin, Inc., New York-Amsterdam, 1969. Notes taken by Hyman Bass. MR 0251042
- Sebastian Casalaina-Martin, Singularities of the Prym theta divisor, Ann. of Math. (2) 170 (2009), no. 1, 162–204. MR 2521114, DOI 10.4007/annals.2009.170.163
- Olivier Debarre, Sur le probleme de Torelli pour les varieties de Prym, Amer. J. Math. 111 (1989), no. 1, 111–134 (French). MR 980302, DOI 10.2307/2374482
- Ron Donagi and Roy Campbell Smith, The structure of the Prym map, Acta Math. 146 (1981), no. 1-2, 25–102. MR 594627, DOI 10.1007/BF02392458
- William Fulton, Intersection theory, Ergebnisse der Mathematik und ihrer Grenzgebiete (3) [Results in Mathematics and Related Areas (3)], vol. 2, Springer-Verlag, Berlin, 1984. MR 732620, DOI 10.1007/978-3-662-02421-8
- M. L. Green, Quadrics of rank four in the ideal of a canonical curve, Invent. Math. 75 (1984), no. 1, 85–104. MR 728141, DOI 10.1007/BF01403092
- Joe Harris, Algebraic geometry, Graduate Texts in Mathematics, vol. 133, Springer-Verlag, New York, 1992. A first course. MR 1182558, DOI 10.1007/978-1-4757-2189-8
- George Kempf, On the geometry of a theorem of Riemann, Ann. of Math. (2) 98 (1973), 178–185. MR 349687, DOI 10.2307/1970910
- G. Kempf, Topics on Riemann surfaces, Univ. Nac. Aut. Mexico, 1973, typed lecture notes, 27 pp.
- George Kempf, Abelian integrals, Monografías del Instituto de Matemáticas [Monographs of the Institute of Mathematics], vol. 13, Universidad Nacional Autónoma de México, México, 1983. MR 743421
- E. Izadi and H. Lange, Counter-examples of high Clifford index to Prym-Torelli, J. Algebraic Geom. 21 (2012), no. 4, 769–787. MR 2957696, DOI 10.1090/S1056-3911-2012-00587-6
- A. Mattuck and A. Mayer, The Riemann-Roch theorem for algebraic curves, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (3) 17 (1963), 223–237. MR 162798
- David Mumford, Prym varieties. I, Contributions to analysis (a collection of papers dedicated to Lipman Bers), Academic Press, New York, 1974, pp. 325–350. MR 0379510
- Sevin 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. MR 480505
- Bernhard Riemann, Collected papers, Kendrick Press, Heber City, UT, 2004. Translated from the 1892 German edition by Roger Baker, Charles Christenson and Henry Orde. MR 2121437
- Igor R. Shafarevich, Basic algebraic geometry. 1, 2nd ed., Springer-Verlag, Berlin, 1994. Varieties in projective space; Translated from the 1988 Russian edition and with notes by Miles Reid. MR 1328833
- V. V. Shokurov, Prym varieties: theory and applications, Izv. Akad. Nauk SSSR Ser. Mat. 47 (1983), no. 4, 785–855 (Russian). MR 712095
- Roy Smith and Robert Varley, Tangent cones to discriminant loci for families of hypersurfaces, Trans. Amer. Math. Soc. 307 (1988), no. 2, 647–674. MR 940221, DOI 10.1090/S0002-9947-1988-0940221-1
- Roy Smith and Robert Varley, Singularity theory applied to $\Theta$-divisors, Algebraic geometry (Chicago, IL, 1989) Lecture Notes in Math., vol. 1479, Springer, Berlin, 1991, pp. 238–257. MR 1181216, DOI 10.1007/BFb0086273
- Roy Smith and Robert Varley, A Riemann singularities theorem for Prym theta divisors, with applications, Pacific J. Math. 201 (2001), no. 2, 479–509. MR 1875904, DOI 10.2140/pjm.2001.201.479
- R. Smith and R. Varley, The Prym Torelli problem: an update and a reformulation as a question in birational geometry, Symposium in Honor of C. H. Clemens (Salt Lake City, UT, 2000) Contemp. Math., vol. 312, Amer. Math. Soc., Providence, RI, 2002, pp. 235–264. MR 1941584, DOI 10.1090/conm/312/05390
- Roy Smith and Robert Varley, A necessary and sufficient condition for Riemann’s singularity theorem to hold on a Prym theta divisor, Compos. Math. 140 (2004), no. 2, 447–458. MR 2027198, DOI 10.1112/S0010437X03000320
- Roy Smith and Robert Varley, The Pfaffian structure defining a Prym theta divisor, The geometry of Riemann surfaces and abelian varieties, Contemp. Math., vol. 397, Amer. Math. Soc., Providence, RI, 2006, pp. 215–236. MR 2218011, DOI 10.1090/conm/397/07475
- A. N. Tjurin, The geometry of the Poincaré divisor of a Prym variety, Izv. Akad. Nauk SSSR Ser. Mat. 39 (1975), no. 5, 1003–1043, 1219 (Russian). MR 0414563
- Robert Varley, Weddle’s surfaces, Humbert’s curves, and a certain $4$-dimensional abelian variety, Amer. J. Math. 108 (1986), no. 4, 931–951. MR 853219, DOI 10.2307/2374519
- Gerald E. Welters, A theorem of Gieseker-Petri type for Prym varieties, Ann. Sci. École Norm. Sup. (4) 18 (1985), no. 4, 671–683. MR 839690, DOI 10.24033/asens.1500
Additional Information
- Roy Smith
- Affiliation: Department of Mathematics, University of Georgia, Athens, Georgia 30602
- MR Author ID: 222535
- Email: roy@math.uga.edu
- Robert Varley
- Affiliation: Department of Mathematics, University of Georgia, Athens, Georgia 30602
- MR Author ID: 222536
- Email: rvarley@math.uga.edu
- Received by editor(s): August 5, 2013
- Received by editor(s) in revised form: October 28, 2014, and May 13, 2015
- Published electronically: December 7, 2016
- © Copyright 2016 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 369 (2017), 3763-3798
- MSC (2010): Primary 14H40; Secondary 14K12
- DOI: https://doi.org/10.1090/tran/6779
- MathSciNet review: 3624392