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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

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$.


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Additional Information

DOI: http://dx.doi.org/10.1090/S0002-9947-1975-0360602-8
PII: S 0002-9947(1975)0360602-8
Keywords: Analytic space, special divisor, Teichmüller surface, moduli
Article copyright: © Copyright 1975 American Mathematical Society