On the Arithmetical Rank of Certain Segre Embeddings

Details On the Arithmetical Rank of Certain Segre Embeddings On the Arithmetical Rank of Certain Segre Embeddings

2010-07-30

arxiv.org/abs/1007.5440v1

Mathematics

Algebraic Geometry

20 pages, to appear on Transaction of the AMS

Scientific paper

We study the number of (set-theoretically) defining equations of Segre products of projective spaces times certain projective hypersurfaces, extending results by Singh and Walther. Meanwhile, we prove some results about the cohomological dimension of certain schemes. In particular, we solve a conjecture of Lyubeznik about an inequality involving the cohomological dimension and the etale cohomological dimension of a scheme, in the characteristic-zero-case and under a smoothness assumption. Furthermore, we show that a relationship between depth and cohomological dimension discovered by Peskine and Szpiro in positive characteristic holds true also in characteristic-zero up to dimension three.

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