抽象除数空间与算术交数

以$cA_n$-奇点为中心的除子收缩计数的扩展摘要

Extended Abstract of Counting Divisorial Contractions with Centre a $cA_n$-Singularity

本文为《代数几何ZAG手册》第149章，综述了发表于《京都大学数学研究所出版物》的、关于以$cA_n$-奇点为中心的除子收缩计数的论文的主要结果。

Divisorial instability and Vojta's Main Conjecture for $\mathbb{Q}$-Fano varieties

We study Diophantine arithmetic properties of birational divisors in conjunction with concepts that surround $\mathrm{K}$-stability for Fano varieties. There is also an interpretation in terms of the barycentres of Newton-Okounkov bodies. Our main results show how the notion of divisorial instability, in the sense of K. Fujita, implies instances of Vojta's Main Conjecture for Fano varieties. A main tool in the proof of these results is an arithmetic form of Cartan's Second Main Theorem that has been obtained by M. Ru and P. Vojta.

Divisorial Cohomology Vanishing on Toric Varieties

This work discusses combinatorial and arithmetic aspects of cohomology vanishing for divisorial sheaves on toric varieties. We obtain a refined variant of the Kawamata-Viehweg theorem which is slightly stronger. Moreover, we prove a new vanishing theorem related to divisors whose inverse is nef and has small Kodaira dimension. Finally, we give a new criterion for divisorial sheaves for being maximal Cohen-Macaulay.

Divisorial linear algebra of normal semigroup rings

We investigate the minimal number of generators $μ$ and the depth of divisorial ideals over normal semigroup rings. Such ideals are defined by the inhomogeneous systems of linear inequalities associated with the support hyperplanes of the semigroup. The main result is that for every bound $C$ there exist, up to isomorphism, only finitely divisorial ideals $I$ such that $μ(I)\le C$. It follows that there exist only finitely many Cohen--Macaulay divisor classes. Moreover we determine the minimal depth of all divisorial ideals and the behaviour of $μ$ and depth in ``arithmetic progressions'' in t…

等分布与算术$Λ$-分布

Equidistribution and arithmetic $Λ$-distributions

本文提出了容许$\mathbb{Z}$-集参数化概率空间族的抽象等分布概念，证明其矩母函数可表为动机Euler乘积，并结合Poonen筛法推广计算了函数域$L$-函数和zeta函数的渐近$Λ$-分布。

Splitting fields of elements in arithmetic groups

We prove that the number of unimodular integral matrices in a norm ball whose characteristic polynomial has Galois group different than the full symmetric group is of strictly lower order of magnitude than the number of all such matrices in the ball, as the radius increases. More generally, we prove a similar result for the Galois groups associated with elements in any connected semisimple linear algebraic group defined and simple over a number field $F$. Our method is based on the abstract large sieve method developed by Kowalski, and the study of Galois groups via reductions modulo primes de…