7,800 research outputs found
A Sheaf Model of the Algebraic Closure
In constructive algebra one cannot in general decide the irreducibility of a
polynomial over a field K. This poses some problems to showing the existence of
the algebraic closure of K. We give a possible constructive interpretation of
the existence of the algebraic closure of a field in characteristic 0 by
building, in a constructive metatheory, a suitable site model where there is
such an algebraic closure. One can then extract computational content from this
model. We give examples of computation based on this model.Comment: In Proceedings CL&C 2014, arXiv:1409.259
Tamagawa numbers of polarized algebraic varieties
Let be an ample metrized invertible sheaf on
a smooth quasi-projective algebraic variety defined over a number field.
Denote by the number of rational points in having -height . We consider the problem of a geometric and arithmetic
interpretation of the asymptotic for as in
connection with recent conjectures of Fujita concerning the Minimal Model
Program for polarized algebraic varieties.
We introduce the notions of -primitive varieties and -primitive fibrations. For -primitive varieties over we
propose a method to define an adelic Tamagawa number which
is a generalization of the Tamagawa number introduced by Peyre for
smooth Fano varieties. Our method allows us to construct Tamagawa numbers for
-Fano varieties with at worst canonical singularities. In a series of
examples of smooth polarized varieties and singular Fano varieties we show that
our Tamagawa numbers express the dependence of the asymptotic of on the choice of -adic metrics on .Comment: 54 pages, minor correction
Order one differential equations on nonisotrivial algebraic curves
In this paper we provide new examples of geometrically trivial strongly
minimal differential algebraic varieties living on nonisotrivial curves over
differentially closed fields of characteristic zero. These are systems whose
solutions only have binary algebraic relations between them. Our technique
involves developing a theory of -forms, and building connections to
deformation theory. This builds on previous work of Buium and Rosen. In our
development, we answer several open questions posed by Rosen and
Hrushovski-Itai
On torsors under elliptic curves and Serre's pro-algebraic structures
Let be a local field with algebraically closed residue field and a
torsor under an elliptic curve over . Let be a proper minimal
regular model of over the ring of integers of and the identity
component of the N\'eron model of . We study the canonical morphism
which extends the biduality isomorphism
on generic fibres. We show that is pro-algebraic in nature with a
construction that recalls Serre's work on local class field theory. Furthermore
we interpret our results in relation to Shafarevich's duality theory for
torsors under abelian varieties.Comment: This paper arises from the confluence and the comparison of the
results contained in the preprints arXiv:1106.1540v2 and arXiv:1005.0462v1 of
the two authors. Final version and to appear in Math. Zeit. We thank the
referee for the very detail review of our pape
K-Theory of non-linear projective toric varieties
By analogy with algebraic geometry, we define a category of non-linear
sheaves (quasi-coherent homotopy-sheaves of topological spaces) on projective
toric varieties and prove a splitting result for its algebraic K-theory,
generalising earlier results for projective spaces. The splitting is expressed
in terms of the number of interior lattice points of dilations of a polytope
associated to the variety. The proof uses combinatorial and geometrical results
on polytopal complexes. The same methods also give an elementary explicit
calculation of the cohomology groups of a projective toric variety over any
commutative ring.Comment: v2: Final version, to appear in "Forum Mathematicum". Minor changes
only, added more cross-referencing and references for toric geometr
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