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A genericity theorem for algebraic stacks and essential dimension of hypersurfaces

Zinovy Reichstein, Angelo Vistoli (2013)

Journal of the European Mathematical Society

We compute the essential dimension of the functors Forms n , d and Hypersurf n , d of equivalence classes of homogeneous polynomials in n variables and hypersurfaces in n 1 , respectively, over any base field k of characteristic 0 . Here two polynomials (or hypersurfaces) over K are considered equivalent if they are related by a linear change of coordinates with coefficients in K . Our proof is based on a new Genericity Theorem for algebraic stacks, which is of independent interest. As another application of the...

ACM bundles on general hypersurfaces in P5 of low degree.

Luca Chiantini, Carlo K. Madonna (2005)

Collectanea Mathematica

In this paper we show that on a general hypersurface of degree r = 3,4,5,6 in P5 a rank 2 vector bundle ε splits if and only if h1ε(n) = h2ε(n) = 0 for all n ∈ Z. Similar results for r = 1,2 were obtained in [15], [16] and [2].

Calculating the Mordell-Weil rank of elliptic threefolds and the cohomology of singular hypersurfaces

Klaus Hulek, Remke Kloosterman (2011)

Annales de l’institut Fourier

In this paper we give a method for calculating the rank of a general elliptic curve over the field of rational functions in two variables. We reduce this problem to calculating the cohomology of a singular hypersurface in a weighted projective 4 -space. We then give a method for calculating the cohomology of a certain class of singular hypersurfaces, extending work of Dimca for the isolated singularity case.

Curves on a ruled cubic surface.

John Brevik, Francesco Mordasini (2003)

Collectanea Mathematica

For the general ruled cubic surface S (with a double line) in P3 = P3 sub k, k any algebraically closed field, we find necessary conditions for which curves on S can be the specialization of a flat family of curves on smooth cubics. In particular, no smooth curve of degree > 10 on S is such a specialization.

Families of hypersurfaces of large degree

Christophe Mourougane (2012)

Journal of the European Mathematical Society

Grauert and Manin showed that a non-isotrivial family of compact complex hyperbolic curves has finitely many sections. We consider a generic moving enough family of high enough degree hypersurfaces in a complex projective space. We show the existence of a strict closed subset of its total space that contains the image of all its sections.

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