Local factorization and monomialization of morphisms

Local monomialization of transcendental extensions

Suppose that f is a dominant morphism from a k-variety X to a k-variety Y, where k is a field of characteristic 0 and v is a valuation of the function field k(X). We allow v to be an arbitary valuation, so it may not be discrete. We prove that there exist sequences of blowups of nonsingular subvarieties from X' to X and from Y' to Y such that X', Y' are nonsingular and X' to Y' is locally a monomial mapping near the center of v. This extends an earlier result of ours (in Asterisque 260) which proves the above result with the restriction that f is generically finite.

函子单项式化与相对主元化中中心的唯一性

Functorial monomialization and uniqueness of centers for relative principalization

本文研究相对对数情形下理想主元化的函子性问题，证明本征映射中Kummer与加权对数中心均可函子式单项式化，从而得到群作用等变且与基局部化相容的特征0相对主元化与对数光滑约化。

Local Strong Factorization of birational maps

The strong factorization conjecture states that a proper birational map between smooth algebraic varieties over a field of characteristic zero can be factored as a sequence of smooth blowups followed by a sequence of smooth blowdowns. We prove a local version of the strong factorization conjecture for toric varieties. Combining this result with the monomialization theorem of S.D.Cutkosky, we obtain a strong factorization theorem for local rings dominated by a valuation.

A refinement of Stein factorization and deformations of surjective morphisms

This paper is concerned with a refinement of the Stein factorization, and with applications to the study of deformations of surjective morphisms. We show that every surjective morphism f:X->Y between normal projective varieties factors canonically via a finite cover of Y that is etale in codimension one. This "maximally etale factorization" is characterized in terms of positivity of the push-forward of the structure sheaf and satisfies a functorial property. It turns out that the maximally etale factorization is stable under deformations, and naturally decomposes an etale cover of the Hom-sche…

Canonical factorization of the quotient morphism for an affine $\mathbb{G}_a$-variety

Working over a ground field of characteristic zero, this paper studies the quotient morphism $π:X\to Y$ for an affine $\mathbb{G}_a$-variety $X$ with affine quotient $Y$. It is shown that the degree modules associated to the $\mathbb{G}_a$-action give a uniquely determined sequence of dominant $\mathbb{G}_a$-equivariant morphisms, $X=X_r\to X_{r-1}\to\cdots\to X_1\to X_0=Y$, where $X_i$ is an affine $\mathbb{G}_a$-variety and $X_{i+1}\to X_i$ is birational for each $i\ge 1$. This is the canonical factorization of $π$. We give an algorithm for finding the degree modules associated to the given …

Canonical factorizations of morphisms of Berkovich curves

We prove that, for certain extensions of valued fields which admit a sensible theory of ramification groups, there exist canonical towers that correspond to the break-points of their Herbrand function. In particular, each of the intermediate field extensions in the tower has a Herbrand function with only one break-point and there is at most one extension with trivial Herbrand function. We apply the result to the setting of finite morphisms of Berkovich curves where we prove the existence of canonical local and global factorization of such morphisms according to their metric properties. Finally…

Normal and Irreducible Adic Spaces, the Openness of Finite Morphisms and a Stein Factorization

We transfer several elementary geometric properties of rigid-analytic spaces to the world of adic spaces, more precisely to the category of adic spaces which are locally of (weakly) finite type over a non-archimedean field. This includes normality, irreducibility (in particular irreducible components) and a Stein factorization theorem. Most notably we show that finite morphisms of adic spaces are open under mild assumptions on the base and target space.

模态映射的分解与到德利涅-穆迪堆栈的通用映射

Factorizations of Moduli Morphisms and Universal Maps to Deligne-Mumford Stacks

本文研究了模态空间映射的分解，构造了简化堆栈结构但保留更多信息的中间堆栈，并证明了在一定条件下存在到满足良好模性质的堆栈的通用映射，展示了该映射本身是有效的模空间映射。

Lyashko-Looijenga态射与Coxeter元的本原因子分解

Lyashko-Looijenga morphisms and primitive factorizations of the Coxeter element

本研究利用平凡化定理规定的LL映射变体，拓展Bessis等人的工作，对良生成复反射群的Coxeter元的本原因子分解进行计数，深化非交叉格的精细计数与结构性质研究。

The canonical embedding of an unramified morphism in an étale morphism

We show that every unramified morphism X->Y has a canonical and universal factorization X->E->Y where the first morphism is a closed embedding and the second is etale (but not separated).

通过因子分解构造纤维化范畴上的幺半群结构

Constructing monoidal structures on fibered categories via factorizations

本研究针对小范畴上纤维化范畴的幺半群结构构造问题，提出利用光滑态射与闭浸入对应的子范畴限制的因子分解方法，证明满足相干条件的限制对可唯一确定整体幺半群结构，并给出适配反常层的逆像替换直像的变体方法。

On the Hitchin morphism for higher dimensional varieties

In this paper, we explore the structure of the Hitchin morphism for higher dimensional varieties. We show that the Hitchin morphism factors through a closed subscheme of the Hitchin base, which is in general a non-linear subspace of lower dimension. We conjecture that the resulting morphism, which we call the spectral data morphism, is surjective. In the course of the proof, we establish connections between the Hitchin morphisms for higher dimensional varieties, the invariant theory of the commuting schemes, and Weyl's polarization theorem. We use the factorization of the Hitchin morphism to c…

proper morphisms 的 Stein 度数

Stein degree of proper morphisms

本文探讨了proper morphisms的Stein度数概念，介绍了其在birational geometry中的应用，并提出了关于Stein度数的有界性问题。

On the birational geometry of toric Fano 4-folds

We announce a factorization result for equivariant birational morphisms between toric 4-folds whose source is Fano: such a morphism is always a composite of blow-ups along smooth invariant centers. Moreover, we show with a counterexample that, differently from the 3-dimensional case, even if both source and target are Fano, the intermediate varieties can not be chosen Fano.

Birational cobordisms and factorization of birational maps

In this paper we develop a Morse-like theory in order to decompose birational maps and morphisms of smooth projective varieties defined over a field of characteristic zero into more elementary steps which are locally étale isomorphic to equivariant flips, blow-ups and blow-downs of toric varieties. A crucial role in the considerations is played by K^*-actions where K is the base field. This paper serves as a basis for proving the weak factorization conjecture on factorization of birational maps in characteristic zero into blow-ups and blow-downs. This is carried out in two subsequent papers, o…

Para-abelian 簇与 Albanese 映射

Para-abelian varieties and Albanese maps

本研究为基域上满足 Stein 分解条件的本征代数空间构造了到 para-abelian 簇的唯一 Albanese 映射，该映射在族情形及群概形作用下保持等变，依赖 para-abelian 簇族与相对 Picard 群 tau 部分可表性等理论。