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


Torsion differentials and deformation

Author: D. S. Rim
Journal: Trans. Amer. Math. Soc. 169 (1972), 257-278
MSC: Primary 14B10; Secondary 14D15, 14M10
MathSciNet review: 0342513
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Abstract: Let $ S$-scheme $ X$ be a Schlessinger deformation of a curve $ {X_0}$ defined over a field $ k$. In §§1 and 2, the dimension of the parameter space $ S$, the relative differentials of $ X$ over $ S$, and the fibres with singularity were studied, in case when $ {X_0}$ is locally complete-intersection. In §3 we show that if $ k$-scheme $ {X_0}$ is a specialization of a smooth $ k$-scheme, then the punctured spectrum $ \operatorname{Spex} ({O_{{X_{0,x}}}})$ has to be connected for every point $ x \in {X_0}$ such that $ \dim {O_{{X_{0,x}}}} \geqslant 2$. In turn we construct a rigid singularity on a surface. In the last section a few conjectures amplifying those of P. Deligne are made.

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PII: S 0002-9947(1972)0342513-4
Article copyright: © Copyright 1972 American Mathematical Society