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Cohomology of certain projective surfaces with low sectional genus and degree. (English) Zbl 0960.14025

Eisenbud, David (ed.), Commutative algebra, algebraic geometry, and computational methods. Proceedings of the conference on algebraic geometry, commutative algebra, and computation, Hanoi, Vietnam, 1996. Singapore: Springer. 173-200 (1999).
The aim of this paper is to prove in any characteristic the joint result of M. Brodmann and W. Vogel [Nagoya Math. J. 131, 109-126 (1993; Zbl 0779.14016)], in characteristic zero, namely: \(\text{reg}(X)\leq\deg(X)-r+3\), where \(X\) is a non-degenerated 2-Buchsbaum surface in \(\mathbb{P}^r\), \(\text{reg}(X)\) is the Castelnuovo regularity and \(\deg(X)\) is the degree. Using the new state of the art, it is essential to prove the above assertion for \(\text{reg}(X)=r+1\) and \(X\) sectionally rational, when the inequality reduces to \(\text{reg}(X)\leq 4\). This is done case by case, according to a classification of the author and P. Schenzel via the Hartshorne-Rao module of a generic hyperplane section of the surface \(X\).
For the entire collection see [Zbl 0921.00043].

MSC:

14M05 Varieties defined by ring conditions (factorial, Cohen-Macaulay, seminormal)
14F20 Étale and other Grothendieck topologies and (co)homologies
14J25 Special surfaces

Citations:

Zbl 0779.14016