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A descent map for curves with totally degenerate semi-stable reduction

Shahed Sharif (2013)

Journal de Théorie des Nombres de Bordeaux

Let K be a local field of residue characteristic p . Let C be a curve over K whose minimal proper regular model has totally degenerate semi-stable reduction. Under certain hypotheses, we compute the prime-to- p rational torsion subgroup on the Jacobian of C . We also determine divisibility of line bundles on C , including rationality of theta characteristics and higher spin structures. These computations utilize arithmetic on the special fiber of C .

A fixed point formula of Lefschetz type in Arakelov geometry II: A residue formula

Kai Köhler, Damien Roessler (2002)

Annales de l’institut Fourier

This is the second of a series of papers dealing with an analog in Arakelov geometry of the holomorphic Lefschetz fixed point formula. We use the main result of the first paper to prove a residue formula "à la Bott" for arithmetic characteristic classes living on arithmetic varieties acted upon by a diagonalisable torus; recent results of Bismut- Goette on the equivariant (Ray-Singer) analytic torsion play a key role in the proof.

A geometric application of Nori’s connectivity theorem

Claire Voisin (2004)

Annali della Scuola Normale Superiore di Pisa - Classe di Scienze

We study (rational) sweeping out of general hypersurfaces by varieties having small moduli spaces. As a consequence, we show that general K -trivial hypersurfaces are not rationally swept out by abelian varieties of dimension at least two. As a corollary, we show that Clemens’ conjecture on the finiteness of rational curves of given degree in a general quintic threefold, and Lang’s conjecture saying that such varieties should be rationally swept-out by abelian varieties, contradict.

A geometric description of Hazama's exceptional classes

Federica Galluzzi (2000)

Bollettino dell'Unione Matematica Italiana

Sia X una varietà abeliana complessa di tipo Mumford. In queste note daremo una descrizione esplicita delle classi eccezionali in B 2 ⁢ X × X trovate da Hazama in [Ha] e le descriveremo geometricamente usando la grassmaniana delle rette di P 7 .

A higher Albanese map for complex threefolds based on a construction by M. Green

Lorenz Schneider (2005)

Fundamenta Mathematicae

We construct a higher Abel-Jacobi map for 0-cycles on complex threefolds and prove that it can be used to describe Mumford's pull-back of a differential form, and that its image is infinite-dimensional in many cases. However, making a certain assumption, we show that it is not always injective.

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