Subscribe to the gold package and get unlimited access to Shamra Academy
Register a new userOn the Bertini regularity theorem for arithmetic varieties
87
0
0.0
(
0
)
Authors
Xiaozong Wang
Ask ChatGPT about the research
No Arabic abstract
Let $mathcal{X}$ be a regular projective arithmetic variety equipped with an ample hermitian line bundle $overline{mathcal{L}}$. We prove that the proportion of global sections $sigma$ with $leftlVert sigma rightrVert_{infty}<1$ of $overline{mathcal{L}}^{otimes d}$ whose divisor does not have a singular point on the fiber $mathcal{X}_p$ over any prime $p<e^{varepsilon d}$ tends to $zeta_{mathcal{X}}(1+dim mathcal{X})^{-1}$ as $drightarrow infty$.
rate research
Read More
We introduce a novel approach to Bertini irreducibility theorems over an arbitrary field, based on random hyperplane slicing over a finite field. Extending a result of Benoist, we prove that for a morphism $phi colon X to mathbb{P}^n$ such that $X$ is geometrically irreducible and the nonempty fibers of $phi$ all have the same dimension, the locus of hyperplanes $H$ such that $phi^{-1} H$ is not geometrically irreducible has dimension at most $operatorname{codim} phi(X)+1$. We give an application to monodromy groups above hyperplane sections.
Because of its ineffectiveness, the usual arithmetic Hilbert-Samuel formula is not applicable in the context of Diophantine Approximation. In order to overcome this difficulty, the present paper presents explicit estimates for arithmetic Hilbert Functions of closed subvarieties in projective space.
We survey some recent work on the geometric Satake of p-adic groups and its applications to some arithmetic problems of Shimura varieties. We reformulate a few constructions appeared in the previous works more conceptually.
Let $F$ be a totally real field in which $p$ is unramified. We study the Goren-Oort stratification of the special fibers of quaternionic Shimura varieties over a place above $p$. We show that each stratum is a $(mathbb{P}^1)^N$-bundle over other quaternionic Shimura varieties (for some appropriate $N$).
We prove an analogue of Kirchhoffs matrix tree theorem for computing the volume of the tropical Prym variety for double covers of metric graphs. We interpret the formula in terms of a semi-canonical decomposition of the tropical Prym variety, via a careful study of the tropical Abel-Prym map. In particular, we show that the map is harmonic, determine its degree at every cell of the decomposition, and prove that its global degree is $2^{g-1}$. Along the way, we use the Ihara zeta function to provide a new proof of the analogous result for finite graphs. As a counterpart, the appendix by Sebastian Casalaina-Martin shows that the degree of the algebraic Abel-Prym map is $2^{g-1}$ as well.
Log in to be able to interact and post comments
comments
Fetching comments
Sign in to be able to follow your search criteria
