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arXiv · 1310.5020

Bertini theorems for singular schemes and nearby cycles in mixed characteristic

Rémi Lodh
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Abstract

We first study hyperplane sections of some singular schemes over a field. We prove a Bertini theorem for the log smoothness of generic hyperplane sections of a large class of log smooth schemes over a log point. We also give an abstract generalization of the usual Bertini theorem for smoothness. We then apply this result to the study of hyperplane sections of regular schemes over a complete discrete valuation ring of characteristic zero with algebraically closed residue field. Lastly, give an exposition of a result of Faltings which states that one can compute the $p$-adic nearby cycles of smooth schemes over a discrete valuation ring of mixed characteristic as Galois cohomology.

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Rémi Lodh. 2014-06-04. Bertini theorems for singular schemes and nearby cycles in mixed characteristic. https://arxiv.org/abs/1310.5020

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