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Transactions of the American Mathematical Society

Published by the American Mathematical Society, the Transactions of the American Mathematical Society (TRAN) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

ISSN 1088-6850 (online) ISSN 0002-9947 (print)

The 2020 MCQ for Transactions of the American Mathematical Society is 1.43.

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On the dimension of varieties of special divisors
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by R. F. Lax PDF
Trans. Amer. Math. Soc. 203 (1975), 141-159 Request permission

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 $f:V_{{T_g}}^{(n)} \to \text {J}$ be the natural relativization over ${T_g}$ of the classical map defined by integrating holomorphic differentials. Let \[ u:{f^\ast }\Omega _{\text {J} /{T_g}}^1 \to \Omega _{V_{{T_g}/{T_g}}^{(n)}}^1\] 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 $G_n^r$.
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Additional Information
  • © Copyright 1975 American Mathematical Society
  • Journal: Trans. Amer. Math. Soc. 203 (1975), 141-159
  • MSC: Primary 14H15; Secondary 14C20, 30A46, 32G15
  • DOI: https://doi.org/10.1090/S0002-9947-1975-0360602-8
  • MathSciNet review: 0360602