Steven Dale Cutkosky
Suppose that $Φ:X\to Y$ is a morphism from a 3 fold to a surface (over an algebraically closed field of characteristic zero). We prove that there exist sequences of blowups of nonsingular subvarieties $X_1\to X$ and $Y_1\to Y$ such that the morphism $Φ_1:X_1\to Y_1$ is a ``Monomial Morphism''. As a corollary, we show that we can make $Φ_1$ a toroidal morphism.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
继续探索
In this paper, local monomialization theorems are proven for analytic morphisms of complex and real analytic spaces. This gives the generalization of the local monomialization theorem for morphisms of algebraic varieties over a field of characteristic zero proven in the author's earlier papers "Local monomialization of morphisms", Asterisque 260 (1999) and "Local monomialization of transcendental extensions", Annales de l'institut Fourier, 1517 - 1586.
We derive extensions of the monomialization theorems for morphisms of varieties in our earlier work. In this note we show that a local monomialization can be found which satisfies stronger local conditions. Some comments are made about how the results of this paper can be used.
We give a short proof of Macintyre's Theorem on Quantifier Elimination for the p-adic numbers, using a version of monomialization that follows directly from the Weak Toroidalization Theorem of Abramovich an Karu (extended to non-closed fields).
Suppose that X to Y is a generically finite map of nonsingular varieties over a field of characteristic zero, and v is a valuation of the function field of X. We prove that it is possible to perform a sequence of monoidal transforms X' to X and Y' to Y so that X' to Y' is a monomial mapping at the center of v. We deduce from this that a birational morphism of nonsingular varieties can be factored along a valuation by a sequence of blowups and blowdowns with nonsingular centers.
We prove a monomialization theorem for mappings in general classes of infinitely differentiable functions that are called quasianalytic. Examples include Denjoy-Carleman classes, the class of $\cC^\infty$ functions definable in a polynomially bounded $o$-minimal structure, as well as the classes of real- or complex analytic functions, and algebraic functions over any field of characteristic zero. The monomialization theorem asserts that a mapping in a quasianalytic class can be transformed to a mapping whose components are monomials with respect to suitable local coordinates, by sequences of s…
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.
In this paper we prove strong toroidalization of birational morphisms of 3-folds. Suppose that f:X\to Y is a birational morphism of nonsingular complete 3-folds, and D_Y, D_X are simple normal crossings divisors on Y and X such that f^{-1}(D_Y)=D_X and D_X contains the singular locus of the morphism f. We prove that there exist morphisms Φ:X_1\to X and Ψ:Y_1\to Y which are products of blow ups of points and nonsingular curves which are supported in the preimage of D_Y and make simple normal crossings with this preimage, such that f_1=Ψ_1^{-1}\circ f\circ Φ_1 is a toroidal morphism. This theore…
We prove that a birational morphism of projective 3-folds, over a field of characteristic zero, can be made toroidal by performing a sequence of blow ups of points and nonsingular curves above the domain and target.
In a work of Costa and Miró-Roig state the following conjecture: Every smooth complete toric Fano variety has a full strongly exceptional collection of line bundles. The goal of this article is to prove it for toric Fano 3-folds.
We give a simpler and more conceptual proof that a morphism from a 3-fold to a surface, over an algebraically closed field of characteristic 0, can be made into a toroidal morphism by sequences of blow ups of nonsingular subvarieties above the domain and range. Our original proof, which is much longer and more complicated, appeared in Springer Lecture Notes in 2002. The final version of this paper is much clearer, thanks to the encouragement of the referee. In particular, Section 5 on global reduction has been completely rewritten. The paper will appear in the Annals of the Fourier Institute.
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.
In this paper, we classify projective toric birational morphisms from Gorenstein toric 3-folds onto the 3-dimensional affine space with relatively ample anti-canonical divisors.
Suppose that $f:X\to Y$ is a dominant morphism of 3-folds over an algebraically closed field of characteristic zero. We prove that there exist sequences of blow ups of points and nonsingular curves $Φ:X_1\to X$ and $Ψ:Y_1\to Y$ such that the induced map $f_1:Y_1\to X_1$ is a toroidal morphism. This extends an earlier proof of the author of this theorem with the extra assumption that $f$ is birational.
The toroidalization conjecture of D. Abramovich, K. Karu, K. Matsuki, and J. Wlodarczyk asks whether any given morphism of nonsingular varieties over an algebraically closed field of characteristic zero can be modified into a toroidal morphism. Following a suggestion by Dale Cutkosky, we define the notion of \emph{locally toroidal} morphisms and ask whether any locally toroidal morphism can be modified into a toroidal morphism. In this paper, we answer the question in the affirmative when the morphism is between any arbitrary variety and a surface.
It is proved that the degree of a morphism from a smooth projective n-fold with Picard number one to a smooth n-quadric is bounded (provided, of course, that n is at least three). Actually it has been proved some years ago, but I have never written down the proof, until J.-M. Hwang suggested, recently, that I do.
We investigate the possibility of embedding minimal abelian surfaces in smooth toric 4-folds with Picard number 2. The existence of such an embedding imposes conditions on the 4-fold, which we partly describe. On the other hand, we exhibit such embeddings in a special case, using a simple description of morphisms into smooth toric varieties.
A new proof of monomialisation from 3-folds to surfaces
本文为三维簇到曲面的态射的单项化提供了新证明，引入对数-Fitting理想并发展新方法，简化了Cutkosky的原始证明。
In this Part II, D(10.2), of D(10), we take D(10.1) (arXiv:1302.2054 [math.AG]) as the foundation to define the notion of $Z$-semistable morphisms from general Azumaya nodal curves, of genus $\ge 2$, with a fundamental module to a projective Calabi-Yau 3-fold and show that the moduli stack of such $Z$-semistable morphisms of a fixed type is compact. This gives us a counter moduli stack to D-strings as the moduli stack of stable maps in Gromov-Witten theory to the fundamental string. It serves and prepares for us the basis toward a new invariant of Calabi-Yau 3-fold that captures soft-D-string …
Some interesting birational morphisms of smooth affine quadric $3$-folds
本文研究复光滑仿射二次三维簇上一族多项式诱导的双有理映射，揭示其上同调双曲性、非代数整数的第二动力学度及周期点增长与代数熵的差异，并结合 Bertini 实验验证。
Let F:X->B be a morphism of varieties in characteristic zero. The problem of semistable reduction of F was stated as a problem in the combinatorics of polyhedral complexes by Abramovich and Karu (alg-geom/9707012). In this paper we solve the combinatorial problem when the relative dimension of F is not bigger than 3.
Suppose $π:W\to S$ is a smooth, proper morphism over a variety $S$ contained as a Zariski open subset in a smooth, complex variety $\bar S$. The goal of this note is to consider the question of when $π$ admits a regular, flat compactification. In other words, when does there exists a flat, proper morphism $\barπ:\overline{W}\to\bar S$ extending $π$ with $\overline{W}$ regular? One interesting recent example of this occurs in the preprint arXiv:1602.05534 of Laza, Sacca and Voisin where $π$ is a family of abelian $5$-folds over a Zariski open subset $S$ of $\bar S=\mathbb{P}^5$. In that paper, …
We prove that the degree of a nonconstant morphism from a smooth projective 3-fold $X$ with Néron-Severi group ${\bf Z}$ to a smooth 3-dimensional quadric is bounded in terms of numerical invariants of $X$. In the special case where $X$ is a 3-dimensional cubic we show that there are no such morphisms. The main tool in the proof is Miyaoka's bound on the number of double points of a surface.
In [L-Y5] (D(6): arXiv:1003.1178 [math.SG]) we introduced the notion of Azumaya $C^{\infty}$-manifolds with a fundamental module and morphisms therefrom to a complex manifold. In the current sequel, we use this notion to give a prototypical definition of supersymmetric D-branes of A-type (i.e. A-branes) -- in an appropriate region of the Wilson's theory-space of string theory -- as special Lagrangian morphisms from such objects with a unitary, minimally flat connection-with-singularities. This merges Donaldson's picture of special Lagrangian submanifolds and the Polchinski-Grothendieck Ansatz …
Classifying Fano 4-folds with a rational fibration onto a 3-fold
本文研究具有到三维簇有理收缩的光滑复 Fano 四维簇，证明非曲面乘积时 Picard 数至多为 9，并分类特殊收缩、构造新高 Picard 数例子。
In a suitable regime of superstring theory, D-branes in a Calabi-Yau space and their most fundamental behaviors can be nicely described mathematically through morphisms from Azumaya spaces with a fundamental module to that Calabi-Yau space. In the earlier work [L-L-S-Y] (D(2): arXiv:0809.2121 [math.AG], with Si Li and Ruifang Song) from the project, we explored this notion for the case of D1-branes (i.e. D-strings) and laid down some basic ingredients toward understanding the notion of D-string world-sheet instantons in this context. In this continuation, D(10), of D(2), we move on to construc…
In this paper we study the Hilbert scheme, Hilb(P), of equidimensional locally Cohen-Macaulay codimension 2 subschemes, with a special look to surfaces in P^4 and 3-folds in P^5, and the Hilbert scheme stratification H_{c} of constant cohomology. For every (X) in Hilb(P) we define a number δ_X in terms of the graded Betti numbers of the homogeneous ideal of X and we prove that 1 + δ_X - \dim_{(X)} H_{c} and 1 + δ_X - \dim T_{c} are CI-biliaison invariants where T_{c} is the tangent space of H_{c} at (X). As a corollary we get a formula for the dimension of any generically smooth component of H…
Higher modularity of elliptic curves over function fields
本文研究函数域上椭圆曲线的高阶模性概念，证明了特征为4的非迷向椭圆曲线的2-模性，借助K3曲面性质完成证明，并得到K3曲面与Kummer曲面有限态射的等价刻画。
扫码添加微信好友，提出您的宝贵建议 👇
💡 备注请填写：网站反馈
发表机构由 AI 从论文文本抽取，如发现错误，欢迎反馈修正建议。