Ramification theory from homotopical point of view, II

从同伦角度看的分歧理论，I

Ramification theory from homotopical point of view, I

本文从同伦角度研究分歧理论，证明了proper映射下 étale sheaf 的推进与特征环推进在p-torsion下的兼容性，采用∞-范畴理论实现。

Comparing commutative and associative unbounded differential graded algebras over Q from homotopical point of view

In this paper we establish a faithfulness result, in a homotopical sense, between a subcategory of the model category of augmented differential graded commutative algebras CDGA and a subcategory of the model category of augmented differential graded algebras DGA over the field of rational numbers $\mathbb{Q}$.

具有指定分支的圆柱面在曲线模空间和$K3$表面塞维里品种上的Brill-Noether locus

Brill-Noether loci of pencils with prescribed ramification on moduli of curves and on Severi varieties on $K3$ surfaces

研究了具有指定分支的圆柱面在曲线模空间和$K3$表面塞维里品种上的Brill-Noether定位，并探讨了其在$K3$表面对应关系和循环中的应用。

On the Jacobian conjecture in characteristic zero

We study the Jacobian conjecture for Keller maps $f:X_0:=\mathbf{A}^n\rightarrow Y_0:=\mathbf{A}^n$ in characteristic $0$ and attempt to prove it. We are quite aware of the fact that many people have tried to prove the Jacobian conjecture before us and hence we stress that this manuscripts is only an attempt. Our approach is to study the finiteness variety $V_f \subset Y_0$ of $f$, the set of points of $Y_0$ over which $f$ fails to be proper. We study a general component $V \subset V_f$ of this set by introducing a suitable representation of $X_0$ which we call the $u-γ$ representation. This v…

代数曲线的几何阿代尔上的 Kummer 理论

Kummer theory over the geometric adeles of an algebraic curve

本文利用几何阿代尔环的 Kummer 理论，通过证明 Picard 群平凡性、建立局部-全局原理及寻找分类不变量，对代数闭域上代数曲线的有限阿贝尔 Galois 覆叠进行了纯代数刻画。