227 research outputs found
Continuity of the radius of convergence of differential equations on -adic analytic curves
This paper deals with connections on -adic analytic curves, in the sense
of Berkovich. The curves must be compact but the connections are allowed to
have a finite number of meromorphic singularities on them. For any choice of a
semistable formal model of the curve, we define an intrinsic notion of
normalized radius of convergence as a function on the curve, with values in
. For a sufficiently refined choice of the semistable model, we prove
continuity and logarithmic concavity of that function. We characterize
\emph{Robba connections}, that is connections whose sheaf of solutions is
constant on any open disk contained in the curve.Comment: 51 pages, 4 figure
Motivic generating series for toric surface singularities
Lejeune-Jalabert and Reguera computed the geometric Poincare series
P_{geom}(T) for toric surface singularities. They raise the question whether
this series equals the arithmetic Poincare series. We prove this equality for a
class of toric varieties including the surfaces, and construct a counterexample
in the general case. We also compute the motivic Igusa Poincare series
Q_{geom}(T) for toric surface singularities, using the change of variables
formula for motivic integrals, thus answering a second question of
Lejeune-Jalabert and Reguera's. The series Q_{geom}(T) contains more
information than the geometric series, since it determines the multiplicity of
the singularity. In some sense, this is the only difference between Q_{geom}(T)
and P_{geom}(T).Comment: 18 page
Local heights of toric varieties over non-archimedean fields
We generalize results about local heights previously proved in the case of
discrete absolute values to arbitrary non-archimedean absolute values of rank
1. First, this is done for the induction formula of Chambert-Loir and
Thuillier. Then we prove the formula of Burgos--Philippon--Sombra for the toric
local height of a proper normal toric variety in this more general setting. We
apply the corresponding formula for Moriwaki's global heights over a finitely
generated field to a fibration which is generically toric. We illustrate the
last result in a natural example where non-discrete non-archimedean absolute
values really matter.Comment: 67 pages. v2: Assumption in Theorem 2.5.8 corrected to support
function; other minor change
Berkovich skeleta and birational geometry
We give a survey of joint work with Mircea Musta\c{t}\u{a} and Chenyang Xu on
the connections between the geometry of Berkovich spaces over the field of
Laurent series and the birational geometry of one-parameter degenerations of
smooth projective varieties. The central objects in our theory are the weight
function and the essential skeleton of the degeneration. We tried to keep the
text self-contained, so that it can serve as an introduction to Berkovich
geometry for birational geometers.Comment: These are expanded lecture notes of a talk at the Simons Symposium on
Non-Archimedean Geometry and Tropical Geometry (March 31-April 6, 2013). They
have been submitted to the conference proceeding
On computing Belyi maps
We survey methods to compute three-point branched covers of the projective
line, also known as Belyi maps. These methods include a direct approach,
involving the solution of a system of polynomial equations, as well as complex
analytic methods, modular forms methods, and p-adic methods. Along the way, we
pose several questions and provide numerous examples.Comment: 57 pages, 3 figures, extensive bibliography; English and French
abstract; revised according to referee's suggestion
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