2,571 research outputs found
Frobenius techniques in birational geometry
This is a survey for the 2015 AMS Summer Institute on Algebraic Geometry
about the Frobenius type techniques recently used extensively in positive
characteristic algebraic geometry. We first explain the basic ideas through
simple versions of the fundamental definitions and statements, and then we
survey most of the recent algebraic geometry results obtained using these
techniques
Zariski Geometries
We characterize the Zariski topologies over an algebraically closed field in
terms of general dimension-theoretic properties. Some applications are given to
complex manifold and to strongly minimal sets.Comment: 9 page
Eulerian digraphs and toric Calabi-Yau varieties
We investigate the structure of a simple class of affine toric Calabi-Yau
varieties that are defined from quiver representations based on finite eulerian
directed graphs (digraphs). The vanishing first Chern class of these varieties
just follows from the characterisation of eulerian digraphs as being connected
with all vertices balanced. Some structure theory is used to show how any
eulerian digraph can be generated by iterating combinations of just a few
canonical graph-theoretic moves. We describe the effect of each of these moves
on the lattice polytopes which encode the toric Calabi-Yau varieties and
illustrate the construction in several examples. We comment on physical
applications of the construction in the context of moduli spaces for
superconformal gauged linear sigma models.Comment: 27 pages, 8 figure
On function field Mordell-Lang and Manin-Mumford
We present a reduction of the function field Mordell-Lang conjecture to the
function field Manin-Mumford conjecture, in all characteristics, via model
theory, but avoiding recourse to the dichotomy theorems for (generalized)
Zariski structures.
In this version 2, the quantifier elimination result in positive
characteristic is extended from simple abelian varieties to all abelian
varieties, completing the main theorem in the positive characteristic case.
In version 3, some corrections are made to the proof of quantifier
elimination in positive characteristic, and the paper is substantially
reorganized.Comment: 21 page
Wolf Barth (1942--2016)
In this article we describe the life and work of Wolf Barth who died on 30th
December 2016. Wolf Barth's contributions to algebraic variety span a wide
range of subjects. His achievements range from what is now called the
Barth-Lefschetz theorems to his fundamental contributions to the theory of
algebraic surfaces and moduli of vector bundles, and include his later work on
algebraic surfaces with many singularities, culminating in the famous Barth
sextic.Comment: accepted for publication in Jahresbericht der Deutschen
Mathematiker-Vereinigung, obituary, 17 pages, 2 figures, 1 phot
Pseudoeffective and nef classes on abelian varieties
The cones of divisors and curves defined by various positivity conditions on
a smooth projective variety have been the subject of a great deal of work in
algebraic geometry, and by now they are quite well understood. However the
analogous cones for cycles of higher codimension and dimension have started to
come into focus only recently. The purpose of this paper is to explore some of
the phenomena that can occur by working out the picture fairly completely in a
couple of simple but non-trivial cases. Specifically, we study cycles of
arbitrary codimension on the self-product of an elliptic curve with complex
multiplication, as well as two dimensional cycles on the product of a very
general abelian surface with itself. Already one finds various non-classical
behavior, for instance nef cycles that fail to be pseudoeffective: this answers
a question raised in 1964 by Grothendieck in correspondence with Mumford. We
also propose a substantial number of open problems for further investigation
- …