Algebra & Number Theory

The degree of the Gauss map of the theta divisor

Giulio Codogni, Samuel Grushevsky, and Edoardo Sernesi

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We study the degree of the Gauss map of the theta divisor of principally polarised complex abelian varieties. Thanks to this analysis, we obtain a bound on the multiplicity of the theta divisor along irreducible components of its singular locus. We spell out this bound in several examples, and we use it to understand the local structure of isolated singular points. We further define a stratification of the moduli space of ppavs by the degree of the Gauss map. In dimension four, we show that this stratification gives a weak solution of the Schottky problem, and we conjecture that this is true in any dimension.

Article information

Algebra Number Theory, Volume 11, Number 4 (2017), 983-1001.

Received: 17 August 2016
Revised: 10 January 2017
Accepted: 11 February 2017
First available in Project Euclid: 16 November 2017

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 14K10: Algebraic moduli, classification [See also 11G15]
Secondary: 14C17: Intersection theory, characteristic classes, intersection multiplicities [See also 13H15] 14H42: Theta functions; Schottky problem [See also 14K25, 32G20]

Gauss map principally polarised abelian varieties Schottky problem V-cycles excess intersection formula


Codogni, Giulio; Grushevsky, Samuel; Sernesi, Edoardo. The degree of the Gauss map of the theta divisor. Algebra Number Theory 11 (2017), no. 4, 983--1001. doi:10.2140/ant.2017.11.983.

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  • A. Andreotti, “On a theorem of Torelli”, Amer. J. Math. 80 (1958), 801–828.
  • E. Arbarello, M. Cornalba, P. A. Griffiths, and J. Harris, Geometry of algebraic curves, I, Grundlehren Math. Wissenschaften 267, Springer, New York, 1985.
  • R. Auffarth, G. P. Pirola, and R. Salvati Manni, “Torsion points on theta divisors”, Proc. Amer. Math. Soc. 145:1 (2017), 89–99.
  • A. Beauville, “Prym varieties and the Schottky problem”, Invent. Math. 41:2 (1977), 149–196.
  • A. Beauville, “Les singularités du diviseur $\Theta $ de la Jacobienne intermédiaire de l'hypersurface cubique dans ${\bf P}\sp{4}$”, pp. 190–208 in Algebraic threefolds (Varenna, Italy, 1981), edited by A. Conte, Lecture Notes in Math. 947, Springer, Berlin, 1982.
  • C. Birkenhake and H. Lange, Complex abelian varieties, 2nd ed., Grundlehren Math. Wissenschaften 302, Springer, Berlin, 2004.
  • C. Ciliberto and G. van der Geer, “The moduli space of abelian varieties and the singularities of the theta divisor”, pp. 61–81 in Surveys in differential geometry, vol. 7, edited by S.-T. Yau, Int. Press, Somerville, MA, 2000.
  • C. H. Clemens and P. A. Griffiths, “The intermediate Jacobian of the cubic threefold”, Ann. of Math. $(2)$ 95 (1972), 281–356.
  • O. Debarre, “Annulation de thêtaconstantes sur les variétés abéliennes de dimension quatre”, C. R. Acad. Sci. Paris Sér. I Math. 305:20 (1987), 885–888.
  • O. Debarre, “Sur les variétés abéliennes dont le diviseur theta est singulier en codimension $3$”, Duke Math. J. 57:1 (1988), 221–273.
  • L. Ein and R. Lazarsfeld, “Singularities of theta divisors and the birational geometry of irregular varieties”, J. Amer. Math. Soc. 10:1 (1997), 243–258.
  • L. Ein, R. Lazarsfeld, and M. Nakamaye, “Zero-estimates, intersection theory, and a theorem of Demailly”, pp. 183–207 in Higher-dimensional complex varieties (Trento, 1994), edited by M. Andreatta and T. Peternell, de Gruyter, Berlin, 1996.
  • H. Flenner, L. O'Carroll, and W. Vogel, Joins and intersections, Springer, Berlin, 1999.
  • W. Fulton, Intersection theory, Ergebnisse der Mathematik $($3$)$ 2, Springer, Berlin, 1984.
  • L. J. van Gastel, “Excess intersections and a correspondence principle”, Invent. Math. 103:1 (1991), 197–222.
  • S. Grushevsky and K. Hulek, “Geometry of theta divisors: a survey”, pp. 361–390 in A celebration of algebraic geometry (Cambridge, 2011), edited by B. Hassett et al., Clay Math. Proc. 18, American Mathematical Society, Providence, RI, 2013.
  • S. Grushevsky and R. Salvati Manni, “Jacobians with a vanishing theta-null in genus 4”, Israel J. Math. 164 (2008), 303–315.
  • J.-i. Igusa, “On the irreducibility of Schottky's divisor”, J. Fac. Sci. Univ. Tokyo Sect. IA Math. 28:3 (1981), 531–545.
  • J. Kollár, Shafarevich maps and automorphic forms, Princeton Univ. Press, 1995.
  • T. Krämer, “Cubic threefolds, Fano surfaces and the monodromy of the Gauss map”, Manuscripta Math. 149:3 (2016), 303–314.
  • H. Matsumura, Commutative ring theory, Cambridge Studies in Advanced Mathematics 8, Cambridge Univ. Press, 1986.
  • D. Mumford, Tata lectures on theta, II, Progress in Mathematics 43, Birkhäuser, Boston, 1984.
  • M. Mustata and M. Popa, “Hodge ideals”, preprint, 2016.
  • M. Nakamaye, “Zero estimates on abelian varieties”, pp. 285–298 in Recent progress in intersection theory (Bologna, 1997), edited by G. Ellingsrud et al., Birkhäuser, Boston, 2000.
  • K. J. Nowak, “A proof of the criterion for multiplicity one”, Univ. Iagel. Acta Math. 35 (1997), 247–249.
  • Z. Ran, “On subvarieties of abelian varieties”, Invent. Math. 62:3 (1981), 459–479.
  • E. Sernesi, Topics on families of projective schemes, Queen's Papers in Pure and Applied Mathematics 73, Queen's University, Kingston, ON, 1986.
  • R. Smith and R. Varley, “Multiplicity $g$ points on theta divisors”, Duke Math. J. 82:2 (1996), 319–326.
  • R. Smith and R. Varley, “Deformations of isolated even double points of corank one”, Proc. Amer. Math. Soc. 140:12 (2012), 4085–4096.
  • R. Varley, “Weddle's surfaces, Humbert's curves, and a certain $4$-dimensional abelian variety”, Amer. J. Math. 108:4 (1986), 931–951.
  • A. Verra, “The degree of the Gauss map for a general Prym theta-divisor”, J. Algebraic Geom. 10:2 (2001), 219–246.
  • W. Vogel, Lectures on results on Bezout's theorem, Tata Institute of Fundamental Research Lectures on Mathematics and Physics 74, Springer, Berlin, 1984.
  • C. Voisin, Théorie de Hodge et géométrie algébrique complexe, Cours Spécialisés 10, Société Mathématique de France, Paris, 2002.
  • H. Żo\l\kadek, The monodromy group, Instytut Mat. Polskiej Akad. Nauk. Mono. Mat. (New Series) 67, Birkhäuser, Basel, 2006.