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Valuations and asymptotic invariants for sequences of ideals

Mattias Jonsson, Mircea Mustaţă (2012)

Annales de l’institut Fourier

We study asymptotic jumping numbers for graded sequences of ideals, and show that every such invariant is computed by a suitable real valuation of the function field. We conjecture that every valuation that computes an asymptotic jumping number is necessarily quasi-monomial. This conjecture holds in dimension two. In general, we reduce it to the case of affine space and to graded sequences of valuation ideals. Along the way, we study the structure of a suitable valuation space.

Vanishing cycles, the generalized Hodge Conjecture and Gröbner bases

Ichiro Shimada (2004)

Banach Center Publications

Let X be a general complete intersection of a given multi-degree in a complex projective space. Suppose that the anti-canonical line bundle of X is ample. Using the cylinder homomorphism associated with the family of complete intersections of a smaller multi-degree contained in X, we prove that the vanishing cycles in the middle homology group of X are represented by topological cycles whose support is contained in a proper Zariski closed subset T of X with certain codimension. In some cases, by...

Vanishing of sections of vector bundles on 0-dimensional schemes

Edoardo Ballico (1999)

Commentationes Mathematicae Universitatis Carolinae

Here we give conditions and examples for the surjectivity or injectivity of the restriction map H 0 ( X , F ) H 0 ( Z , F | Z ) , where X is a projective variety, F is a vector bundle on X and Z is a “general” 0 -dimensional subscheme of X , Z union of general “fat points”.

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