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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
DOI: https://doi.org/10.1090/S0002-9947-1972-0342513-4
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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Article copyright: © Copyright 1972 American Mathematical Society