Asymptotic Purity for Very General Hypersurfaces of P^n x P^n of Bidegree (k,k)

Details Asymptotic Purity for Very General Hypersurfaces of P^n x P^n of Bidegree (k,k) Asymptotic Purity for Very General Hypersurfaces of P^n x P^n of Bidegree (k,k)

2011-04-05

arxiv.org/abs/1104.0726v1

Mathematics

Algebraic Geometry

11 pages

Scientific paper

For a complex irreducible projective variety, the volume function and the higher asymptotic cohomological functions have proven to be useful in understanding the positivity of divisors as well as other geometric properties of the variety. In this paper, we study the vanishing properties of these functions on specific hypersurfaces of P^n \times P^n. In particular, we show that very general hypersurfaces of bidegree (k,k) obey a very strong vanishing property, which we define as asymptotic purity: at most one asymptotic cohomological function is nonzero for each divisor. This provides evidence for a conjecture of Bogomolov and also suggests some general conditions for asymptotic purity.

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