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On the dimension of varieties of special divisors


Author: R. F. Lax
Journal: Trans. Amer. Math. Soc. 203 (1975), 141-159
MSC: Primary 14H15; Secondary 14C20, 30A46, 32G15
MathSciNet review: 0360602
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Abstract: Let $ {T_g}$ denote the Teichmüller space and let $ V$ denote the universal family of Teichmüller surfaces of genus $ g$ Let $ V_{{T_g}}^{(n)}$ denote the $ n$th symmetric product of $ V$ over $ {T_g}$ and let $ J$ denote the family of Jacobians over $ {T_g}$. Let $ f:V_{{T_g}}^{(n)} \to$   J$ $ be the natural relativization over $ {T_g}$ of the classical map defined by integrating holomorphic differentials. Let

$\displaystyle u:{f^\ast }\Omega _{\text{J} /{T_g}}^1 \to \Omega _{V_{{T_g}/{T_g}}^{(n)}}^1$

be the map induced by $ f$. We define $ G_n^r$ to be the analytic subspace of $ V_{{T_g}}^{(n)}$ defined by the vanishing of $ { \wedge ^{n - r + 1}}u$.

Put $ \tau = (r + 1)(n - r) - rg$. We show that $ G_n^1 - G_n^2$, if nonempty, is smooth of pure dimension $ 3g - 3 + \tau + 1$. From this result, we may conclude that, for a generic curve $ X$, the fiber of $ G_n^1 - G_n^2$ over the module point of $ X$, if nonempty, is smooth of pure dimension $ \tau + 1$, a classical assertion.

Variational formulas due to Schiffer and Spencer and Rauch are employed in the study of $ G_n^r$.


References [Enhancements On Off] (What's this?)

  • [1] Aldo Andreotti, On a theorem of Torelli, Amer. J. Math. 80 (1958), 801–828. MR 0102518
  • [1a] 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. MR 0220740
  • [2] A. Brill and M. Noether, Über die algebraischen Funktionen und ihre Anwendung in der Geometrie, Math. Ann. 7 (1874).
  • [3] Hershel M. Farkas, Special divisors and analytic subloci of Teichmueller space, Amer. J. Math. 88 (1966), 881–901. MR 0213546
  • [4] Hans Grauert, Ein Theorem der analytischen Garbentheorie und die Modulräume komplexer Strukturen, Inst. Hautes Études Sci. Publ. Math. 5 (1960), 64 (German). MR 0121814
  • [5] A. Grothendieck, Exposés in Séminaire Cartan, 1960/61, Secrétariat mathématique, Paris.
  • [6] R. C. Gunning, Lectures on Riemann surfaces, Jacobi varieties, Princeton University Press, Princeton, N.J.; University of Tokyo Press, Tokyo, 1972. Mathematical Notes, No. 12. MR 0357407
  • [7] Birger Iversen, Linear determinants with applications to the Picard scheme of a family of algebraic curves, Lecture Notes in Mathematics, Vol. 174, Springer-Verlag, Berlin-New York, 1970. MR 0292835
  • [8] G. Kempf, Schubert methods with an application to algebraic curves, Stichting Mathematisch Centrum, Amsterdam, 1971.
  • [9] G. Kempf and D. Laksov, The determinantal formula of Schubert calculus, Acta Math. 132 (1974), 153–162. MR 0338006
  • [10] Steven L. Kleiman and Dan Laksov, On the existence of special divisors, Amer. J. Math. 94 (1972), 431–436. MR 0323792
  • [11] Steven L. Kleiman and Dan Laksov, Another proof of the existence of special divisors, Acta Math. 132 (1974), 163–176. MR 0357398
  • [12] Henrik H. Martens, On the varieties of special divisors on a curve, J. Reine Angew. Math. 227 (1967), 111–120. MR 0215847
  • [13] Henrik H. Martens, Varieties of special divisors on a curve. II, J. Reine Angew. Math. 233 (1968), 89–100. MR 0241420
  • [14] A. Mattuck and A. Mayer, The Riemann-Roch theorem for algebraic curves, Ann. Scuola Norm. Sup. Pisa (3) 17 (1963), 223–237. MR 0162798
  • [15] Alan Mayer, Rauch’s variational formula and the heat equation, Math. Ann. 181 (1969), 53–59. MR 0248336
  • [16] Theodor Meis, Die minimale Blätterzahl der Konkretisierungen einer kompakten Riemannschen Fläche, Schr. Math. Inst. Univ. Münster No. 16 (1960), 61 (German). MR 0147643
  • [17] Charles Patt, Variations of Teichmueller and Torelli surfaces, J. Analyse Math. 11 (1963), 221–247. MR 0160894
  • [18] H. E. Rauch, Weierstrass points, branch points, and moduli of Riemann surfaces, Comm. Pure Appl. Math. 12 (1959), 543–560. MR 0110798
  • [19] Menahem Schiffer and Donald C. Spencer, Functionals of finite Riemann surfaces, Princeton University Press, Princeton, N. J., 1954. MR 0065652
  • [20] Beniamino Segre, Sui moduli delle curve poligonali, e sopra un complemento al teorema di esistenza di Reimann, Math. Ann. 100 (1928), no. 1, 537–551 (Italian). MR 1512501, 10.1007/BF01448862
  • [21] F. Severi and E. Löffler, Vorlesungen über algebraischen Geometrie, Teubner, Leipzig, 1921.
  • [22] -, Sul teorema di esistenza di Riemann, Rend. Circ. Mat. Palermo 46 (1922).
  • [23] George Springer, Introduction to Riemann surfaces, Addison-Wesley Publishing Company, Inc., Reading, Mass., 1957. MR 0092855

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Additional Information

DOI: https://doi.org/10.1090/S0002-9947-1975-0360602-8
Keywords: Analytic space, special divisor, Teichmüller surface, moduli
Article copyright: © Copyright 1975 American Mathematical Society