Decomposition of log crepant birational morphisms between log terminal surfaces

log Calabi-Yau 3-折叠的双有理复杂度

Birational complexity of log Calabi-Yau 3-folds

本文研究log Calabi-Yau 3-折叠的双有理复杂度，证明其取值于{0,2,3}，并构造具有到射影空间收缩的crepant双有理模型，通过标准$\mathbb{P}^1$-链的几何刻画实现。

Toroidalization of birational morphisms of 3-folds

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.

Poisson deformations and birational geometry

Let π: Y -> X be a crepant projective resolution of an affine symplectic variety X with a good C^*-action. We interpret the second cohomology H^2(Y, C) in two ways. First, H^2(Y, C) is the Picard group of Y tensorised with C. By the ample cones of different crepant resolutions of X, there is a natural chamber structure in H^2(Y, C). The second interpretation of H^2(Y, C) is the base space of the universal Poisson deformation $\mathcal Y$ of Y. Let D \subset H^2(Y, C) be the locus where the corresponding Poisson varieties are not affine. Then D is the union of finite number of hyperplanes, whic…

Birational morphisms and Poisson moduli spaces

We study birational morphisms between smooth projective surfaces that respect a given Poisson structure, with particular attention to induced birational maps between the (Poisson) moduli spaces of sheaves on those surfaces. In particular, to any birational morphism, we associate a corresponding "minimal lift" operation on sheaves of homological dimension <=1, and study its properties. In particular, we show that minimal lift induces a stratification of the moduli space of simple sheaves on the codomain by open subspaces of the moduli space of simple sheaves on the domain, compatibly with the i…

Numerical properties of exceptional divisors of birational morphisms of smooth surfaces

We make a very detailed analysis of the numerical properties of effective divisors whose support is contained in the exceptional locus of a birational morphism of smooth projective surfaces. As an application we extend Miyaoka's inequality on the number of canonical singularities on a projective normal surface with non-negative Kodaira dimension to the non minimal case, obtaining a slightly better result than known extensions by Megyesi and Langer.

The Cohomological Crepant Resolution Conjecture for the Hilbert-Chow morphisms

In this paper, we prove that Ruan's Cohomological Crepant Resolution Conjecture holds for the Hilbert-Chow morphisms. There are two main ideas in the proof. The first one is to use the representation theoretic approach proposed in [QW] which involves vertex operator techniques. The second is to prove certain universality structures about the 3-pointed genus-0 extremal Gromov-Witten invariants of the Hilbert schemes by using the indexing techniques from [LiJ], the product formula from [Beh2] and the co-section localization from [KL1, KL2, LL]. We then reduce Ruan's Conjecture from the case of a…

Wall-Crossings in Toric Gromov-Witten Theory II: Local Examples

In this paper we analyze six examples of birational transformations between toric orbifolds: three crepant resolutions, two crepant partial resolutions, and a flop. We study the effect of these transformations on genus-zero Gromov-Witten invariants, proving the Coates-Corti-Iritani-Tseng/Ruan form of the Crepant Resolution Conjecture in each case. Our results suggest that this form of the Crepant Resolution Conjecture may also hold for more general crepant birational transformations. They also suggest that Ruan's original Crepant Resolution Conjecture should be modified, by including appropria…

On the Crepant Resolution Conjecture in the Local Case

In this paper we analyze four examples of birational transformations between local Calabi-Yau 3-folds: two crepant resolutions, a crepant partial resolution, and a flop. We study the effect of these transformations on genus-zero Gromov-Witten invariants, proving the Coates-Corti-Iritani-Tseng/Ruan form of the Crepant Resolution Conjecture in each case. Our results suggest that this form of the Crepant Resolution Conjecture may also hold for more general crepant birational transformations. They also suggest that Ruan's original Crepant Resolution Conjecture should be modified, by including appr…

On a vanishing theorem for birational morphisms of threefolds in positive and mixed characteristics

We prove a relative Kawamata Viehweg vanishing type theorem for birational morphisms. We use this to prove a Grauert Riemenschneider theorem over log canonical threefolds without zero dimensional log canonical centers, in residue characteristic p greater than five. In large enough residue characteristics, we prove a Grauert Riemenschneider theorem over threefold log canonical singularities with standard coefficients. These vanishing theorems can also be used to study the depth of log canonical singularities at non log canonical centers, as well as the singularities of the log canonical centers…

Dynamical Mordell-Lang conjecture for birational polynomial morphisms on $\mathbb{A}^2$

We prove the dynamical Mordell-Lang conjecture for birational polynomial morphisms on $\mathbb{A}^2$.

Toric Fano varieties and birational morphisms

In this paper we study smooth toric Fano varieties using primitive relations and toric Mori theory. We show that for any irreducible invariant divisor D in a toric Fano variety X, we have $0\leqρ_X-ρ_D\leq 3$, for the difference of the Picard numbers of X and D. Moreover, if $ρ_X-ρ_D>0$ (with some additional hypotheses if $ρ_X-ρ_D=1$), we give an explicit birational description of X. Using this result, we show that when dim X=5, we have $ρ_X\leq 9$. In the second part of the paper, we study equivariant birational morphisms f whose source is Fano. We give some general results, and in dimension …

有理奇点与 $q$-双有理态射

Rational singularities and $q$-birational morphism

本文推广了任意秩为1的反射层的奇点概念，将其与[Kov11]中的有理奇点联系起来，并证明了有理奇点标准事实的推广；通过定义非有理点集，引入$(B_{q+1})$作为Serre的$(S_{q+1})$的对偶概念，并证明了关于$q$-双有理态射的定理。

关于带有环面作用的log Calabi-Yau对的Albanese映射的birational isotrivial性

On the birational isotriviality of the Albanese morphism of a log Calabi-Yau pair with a torus action

本文研究了带有环面作用的log Calabi-Yau对的Albanese映射的birational isotrivial性，证明了在特定条件下，Albanese映射的两个一般纤维是birationally等价的。

Birational geometry of quiver varieties and other GIT quotients

We prove that all projective crepant resolutions of Nakajima quiver varieties satisfying natural conditions are also Nakajima quiver varieties. More generally, we classify the small birational models of many Geometric Invariant Theory (GIT) quotients by introducing a sufficient condition for the GIT quotient of an affine variety $V$ by the action of a reductive group $G$ to be a relative Mori Dream Space. Two surprising examples illustrate that our new condition is optimal. When the condition holds, we show that the linearisation map identifies a region of the GIT fan with the Mori chamber dec…

Piecewise morphisms of birational foliated varieties

This paper has been withdrawn by the author due to a crucial error.

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.

量子刻面几何中的准合理映射

Birational morphisms in quantum toric geometry

本文研究量子刻面几何中由加权吹胀和边界关系诱导的合理映射，探讨其在刻面堆中的应用与特性。

On relative birational geometry and Nagata's compactification

In 2011, the first author introduced (relative) Riemann-Zariski spaces corresponding to a morphism of schemes and established their basic properties. In this paper we clarify that theory and extend it to morphisms between algebraic spaces. As an application, a new proof of Nagata's compactification theorem for algebraic spaces is obtained.

Poisson structures and birational morphisms associated with bundles on elliptic curves

In this paper we define a Poisson structure on some moduli spaces related to principal G-bundles on elliptic curves, the simplest example being the moduli space of stable pairs: a vector bundle and its global section. We also study birational morphisms between projective spaces appearing as such moduli spaces.

$F$-signature under birational morphisms

We study $F$-signature under proper birational morphisms $π: Y \to X$, showing that $F$-signature strictly increases for small morphisms or if $ K_Y \geq π^*K_X$. In certain cases, we can even show that the $F$-signature of $Y$ is at least twice as that of $X$. We also provide examples of $F$-signature dropping and Hilbert-Kunz multiplicity increasing under birational maps without these hypotheses.

极值复杂度与圆锥纤维化

Birational complexity and conic fibrations

本文研究了log Calabi-Yau对的极值复杂度与圆锥纤维化的关系，并引入了新的复杂度度量标准，探讨了其在birational超刚性、Fano超曲面等领域的应用。

Birational morphisms of the plane

Let A^2 be the affine plane over a field K of characteristic 0. Birational morphisms of A^2 are mappings A^2 \to A^2 given by polynomial mappings ϕof the polynomial algebra K[x,y] such that for the quotient fields, one has K(ϕ(x), ϕ(y)) = K(x,y). Polynomial automorphisms are obvious examples of such mappings. Another obvious example is the mapping τ_x given by x \to x, y \to xy. For a while, it was an open question whether every birational morphism is a product of polynomial automorphisms and copies of τ_x. This question was answered in the negative by P. Russell (in an informal communication)…

Three-dimensional toric morphisms with anti-nef canonical divisors

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.

Induced nilpotent orbits and birational geometry

In general, a nilpotent orbit closure in a complex simple Lie algebra \g, does not have a crepant resolution. But, it always has a Q-factorial terminalization by the minimal model program. According to B. Fu, a nilpotent orbit closure has a crepant resolution only when it is a Richardson orbit, and the resolution is obtained as a Springer map for it. In this paper, we shall generalize this result to Q-factorial terminalizations when \g$ is classical. Here, the induced orbits play an important role instead of Richardson orbits.

复杂度为零且指数任意的对数 Calabi--Yau 对

Log Calabi--Yau pairs of complexity zero and arbitrary index

本文刻画了复杂度为零且指数任意的对数 Calabi--Yau 对，并证明双有理复杂度为零的对数 Calabi--Yau 对存在 crepant 双有理模型，即广义 Bott 塔。

Cox rings of morphisms and resolution of singularities

We extend the Cox-Hu-Keel construction of the Cox rings to any proper birational morphisms of normal noetherian schemes. It allows the representation of any proper birational morphism by a map of schemes with mild singularities with torus actions. In a particular case, the notion generalizes the combinatorial construction of Satriano and the recent construction of multiple weighted blow-ups on Artin-stacks by Abramovich-Quek. The latter can be viewed as an extension of stack theoretic blow-ups by Abramovich, Temkin, and Wlodarczyk, a similar construction of McQuillan and the author's recent co…

Functorial factorization of birational maps for qe schemes in characteristic 0

We prove functorial weak factorization of projective birational morphisms of regular quasi-excellent schemes in characteristic 0 broadly based on the existing line of proof for varieties. From this general functorial statement we deduce factorization results for algebraic stacks, formal schemes, complex analytic germs, Berkovich analytic and rigid analytic spaces.

Birational contraction of genus two tails in the moduli space of genus four curves I

We show that for $α\in (2/3, 7/10)$, the log canonical model $\bar M_4(α)$ of the pair $(\bar M_4, αδ)$ is isomorphic to the moduli space $\bar M_4^{hs}$ of h-semistable curves, and that there is a birational morphism $Ξ: \bar M_4^{hs} \to \bar M_4(2/3)$ that contracts the locus of curves $C_1\cup_p C_2$ consisting of genus two curves meeting in a node $p$ such that $p$ is a Weierstrass point of $C_1$ or $C_2$. To obtain this morphism, we construct a compact moduli space $\bar M_{2,1}^{hs}$ of pointed genus two curves that have nodes, ordinary cusps and tacnodes as singularity, and prove that …

关于锥面纤维的理性和自映射群的评论

Remarks on the group of birational selfmaps of a conic fibration

研究锥面纤维的理性和自映射群，证明其存在到不可数多个$\mathbb Z/2\IZ$副本直接和的满射映射。

正则局部环的纯子环、自同态环与Frobenius态射

Pure subrings of regular local rings, endomorphism rings and Frobenius morphisms

本文研究正则局部环纯子环的自同态环，证明其在极大Cohen-Macaulay条件下为非交换crepant消解，并利用Frobenius态射构造Morita等价的另一消解，同时探讨非交换Frobenius态射与整体维数有限性的关系，并应用于wild商奇点的强F-正则性判定。

可积族中理性有界的叶体积

Volumes of foliations birationally bounded by algebraically integrable families

研究证明了由代数可积族理性有界所限定的log canonical叶体积集合满足DCC，并建立了弱半稳定映射家族的相对log canonical体积的变形不变性。

On Picard bundles over Prym varieties

We study the birationality (onto its image) of the Abel-Prym morphism associated with a Prym-Tuyrin variety. We use such result to prove that Picard bundles over Prym varieties are simple and moreover they are stable when the Abel-Prym morphism is not birational. As a consequence we obtain that Picard bundles over moduli spaces of stable vector bundles with fixed determinant are simple. We prove that Picard bundles over moduli spaces of rank 2 vector bundles on curves of genus 2 bundles are stable.

The classification of surfaces with $p_g=0$, $K^2=6$ and non birational bicanonical map

Let $S$ be a minimal surface of general type with $p_g=0$ and $K^2=6$, such that its bicanonical map $\fie\colon S\to\pp^6$ is not birational. The map $\fie$ is a morphism of degree $\le 4$ onto a surface. The case of $°\fie=4$ is completely characterized in [Topology, {\bf 40} (5) (2001), 977--991] and the present paper completes the classification of these surfaces. It is proven that the degree of $\fie$ cannot be equal to 3, and the geometry of surfaces with $°\fie=2$ is analysed in detail. The last section contains three examples of such surfaces, two of which appear to be new.

On the image of Hitchin morphism for algebraic surfaces: The case ${\rm GL}_n$

The Hitchin morphism is a map from the moduli space of Higgs bundles $\mathscr{M}_X$ to the Hitchin base $\mathscr{B}_X$, where $X$ is a smooth projective variety. When $X$ has dimension at least two, this morphism is not surjective in general. Recently, Chen-Ngô introduced a closed subscheme $\mathscr{A}_X$ of $\mathscr{B}_X$, which is called the space of spectral data. They proved that the Hitchin morphism factors through $\mathscr{A}_X$ and conjectured that $\mathscr{A}_X$ is the image of the Hitchin morphism. We prove that when $X$ is a smooth projective surface, this conjecture is true fo…

Blowups and fibers of morphisms

Our object of study is a rational map Psi from projective s-1 space to projective n-1 space defined by homogeneous forms g1,...,gn, of the same degree d, in the homogeneous coordinate ring R=k[x1,...,xs] of projective s-1 space. Our goal is to relate properties of Psi, of the homogeneous coordinate ring A=k[g1,...,gn] of the variety parametrized by Psi, and of the Rees algebra R[It], the bihomogeneous coordinate ring of the graph of Psi. For a regular map Psi, for instance, we prove that R[It] satisfies Serre's condition R_i, for some positive i, if and only if A satisfies R_{i-1} and Psi is b…

基本群概形的双有理不变性

The Birational Invariance Of Fundamental Group Schemes

本文建立了Tannakian范畴间自然同态成为同构的统一准则，并证明在完美域上光滑射影簇的双有理映射下，多种基本群概形（如S、N、EN等）均自然同构。

Note on quasi-polarized canonical Calabi-Yau threefolds

Let $(X,L)$ be a quasi-polarized canonical Calabi-Yau threefold. In this note, we show that $\vert mL\vert$ is basepoint free for $m\geq 4$. Moreover, if the morphism $Φ_{\vert 4L\vert}$ is not birational onto its image and $h^0(X,L)\geq 2$, then $L^3=1$. As an application, if $Y$ is a $n$-dimensional Fano manifold such that $-K_Y=(n-3)H$ for some ample divisor $H$, then $\vert mH\vert$ is basepoint free for $m\geq 4$ and if the morphism $Φ_{\vert 4H\vert}$ is not birational onto its image, then $Y$ is either a weighted hypersurface of degree $10$ in the weighted projective space $\mathbb{P}(1…

光滑仿射二次三维簇的一些有趣双有理态射

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。

Explicit biregular/birational geometry of affine threefolds: completions of A^3 into del Pezzo fibrations and Mori conic bundles

We study certain pencils of del Pezzo surfaces generated by a smooth del Pezzo surface S of degree less or equal to 3 anti-canonically embedded into a weighted projective space P and an appropriate multiple of a hyperplane H. Our main observation is that every minimal model program relative to the morphism lifting such pencil on a suitable resolution of its indeterminacies preserves the open subset P H â A^3. As an application, we obtain projective completions of A^3 into del Pezzo fibrations over P^1 of every degree less or equal to 4. We also obtain completions of A^3 into Mori conic bundles…

Weak toroidalization over non-closed fields

We prove that any dominant morphism of algebraic varieties over a field k of characteristic zero can be transformed into a toroidal (hence monomial) morphism by projective birational modifications of source and target. This was previously proved by the first and third author when k is algebraically closed. Moreover we show that certain additional requirements can be satisfied.

On the birational geometry of Fano 4-folds

We study the birational geometry of a Fano 4-fold X from the point of view of Mori dream spaces; more precisely, we study rational contractions of X. Here a rational contraction is a rational map f: X-->Y, where Y is normal and projective, which factors as a finite sequence of flips, followed by a surjective morphism with connected fibers. Such f is called elementary if the difference of the Picard numbers of X and Y is 1. We first give a characterization of non-movable prime divisors in X, when X has Picard number at least 6; this is related to the study of birational and divisorial elementar…

Higher direct images of the structure sheaf in positive characteristic

We prove vanishing of the higher direct images of the structure (and the canonical) sheaf for a proper birational morphism with source a smooth variety and target the quotient of a smooth variety by a finite group of order prime to the characteristic of the ground field. We also show that for smooth projective varieties the cohomology of the structure sheaf is a birational invariant. These results are well-known in characteristic zero.

On the exceptional locus of the birational projections of normal surface singularity into a plane

Given a normal surface singularity $(X, Q)$ and a birational morphism to a non- singular surface $π: X \to S$, we investigate the local geometry of the exceptional divisor $L$ of $π$. We prove that the dimension of the tangent space to $L$ at $Q$ equals the number of exceptional components meeting at $Q$. Consequences relative to the existence of such birational projections contracting a prescribed number of irreducible curves are deduced. A new characterization of minimal singularities is obtained in these terms.

Vanishing of the higher direct images of the structure sheaf

We prove that the higher direct images of the structure sheaf under a birational and projective morphism between excellent and regular schemes vanish.

Non-locally-free locus of O'Grady's ten dimensional example

We give a completely explicit description of the fibers of the natural birational morphism from O'Grady's ten dimensional singular moduli space of sheaves on a K3 surface to the corresponding Donaldson-Uhlenbeck compactification.

曲面上的稳定性条件与曲线收缩

Stability conditions on surfaces and contractions of curves

研究源自双有理态射的光滑曲面上的Bridgeland稳定性条件，在ADE或循环商奇点下构造预稳定性条件并证明支持性质，同时证明此类态射不能收缩正亏格的光滑曲线。

Morphisms of Projective Varieties from the viewpoint of Minimal Model Theory

In this Lecture Notes we present, in a sufficiently self contained way, our contributions and interests in the field of Minimal Model Theory. We study Fano-Mori spaces, both from the biregular and the birational point of view. For the former we recall and develop Kawamata's Base Point Free technique and some of Mori's deformation arguments. For the latter we lean on Sarkisov and #-Minimal Model Programs. In writing these notes we want to give our point of view on this area of research. We are not trying to give a treatment of the whole subject. These notes collect some topics we presented in t…

Bounds for log canonical thresholds with applications to birational rigidity

We use intersection theory, degeneration techniques and jet schemes to study log canonical thresholds. Our first result gives a lower bound for the log canonical threshold of a pair in terms of the log canonical threshold of the image by a suitable smooth morphism. This in turn is based on an inequality relating the log canonical threshold and the Samuel multiplicity, generalizing our previous result from math.AG/0205171. We then give a lower bound for the log canonical threshold of an affine scheme defined by homogeneous equations of the same degree in terms of the dimension of the non log te…

On the birational geometry of Hilbert schemes of points and Severi divisors

We study the birational geometry of Hilbert schemes of points on non-minimal surfaces. In particular, we study the weak Lefschetz Principle in the context of birational geometry. We focus on the interaction of the stable base locus decomposition (SBLD) of the cones of effective divisors of $X^{[n]}$ and $Y^{[n]}$, when there is a birational morphism $f:X\rightarrow Y$ between surfaces. In this setting, $N^1(Y^{[n]})$ embeds in $N^1(X^{[n]})$, and we ask if the restriction of the stable base locus decomposition of $N^1(X^{[n]})$ yields the respective decomposition in $N^1(Y^{[n]})$ $i.e.$, if t…

A compactification of $\mathcal M_3$ via $K3$ surfaces

S. Kondō defined a birational period map from the moduli space of genus three curves to a moduli space of degree four polarized K3 surfaces. In this paper we extend the period map to a surjective morphism on a suitable compactification of $\mathcal M_3$ and describe its geometry.

射影曲面上的sheaf稳定对与birational几何

Sheaf stable pairs on projective surfaces and birational geometry

本文研究了射影曲面上更高秩稳定对的模空间，并利用极小模型的birational几何分析其与基曲面的联系。

Cox rings of rational surfaces and redundant blow-ups

We prove that the redundant blow-up preserves the finite generation of the Cox ring of a rational surface under a suitable assumption, and we study the birational structure of Mori dream rational surfaces via redundant blow-ups. It turns out that the redundant blow-up completely characterizes birational morphisms of Mori dream rational surfaces with anticanonical Iitaka dimension $0$. As an application, we construct new Mori dream rational surfaces with anticanonical Iitaka dimension $0$ and $-\infty$ of arbitrarily large Picard number.

双有理局部化中的维数过滤

Dimension filtrations in birational localisation

本文研究Kahn-Sujatha关于双有理局部化中维数过滤函子忠实性的问题，证明仅当n=0时自然函子忠实，并给出维数阈值条件。

A fibered power theorem for pairs of log general type

Let $f: (X,D) \to B$ be a stably family with log canonical general fiber. We prove that, after a birational modification of the base $\tilde{B} \to B$, there is a morphism from a high fibered power of the family to a pair of log general type. If in addition the general fiber is openly canonical, then there is a morphism from a high fibered power of the original family to a pair openly of log general type.