伴随有理曲面的一个不等式

泛中心化子的上同调 I：伴随群情形

Cohomology of the universal centralizers I: the adjoint group case

本研究计算了复伴随型半单群对应泛中心化子$J_G$的有理上同调，发现其拓扑虽随群秩升高变复杂，但有理上同调与单点一致，后续将推广至一般半单群并证明其有理上同调具纯Hodge结构。

有理极化Hodge结构的Mumford--Tate群的分类

Classification of the Mumford--Tate Groups of Rational Polarizable Hodge Structures

本文通过分解与极化有理Hodge结构相伴的pro-代数群G，将其商、导出群及伴随群分别刻画为Serre群、单连通覆盖及特定单代数群的乘积，从而分类了Mumford--Tate群。

正特征正规曲面上的消失定理与伴随线性系统

Vanishing theorems and adjoint linear systems on normal surfaces in positive characteristic

本文证明了正特征曲面上大类的Kawamata-Viehweg消失定理，并由此建立Reider型定理与态射延拓定理，进而用有理函数刻画任意基域上平面曲线的非奇异有理点。

托利安幅度与通用伴随

Toric Amplitudes and Universal Adjoints

本文研究了托利安幅度的代数性质及其与超曲面几何的关系，通过双体积解释揭示其与Warren伴随的联系。

曲线Gorenstein伴随理想的局部到全局算法

Local to global algorithms for the Gorenstein adjoint ideal of a curve

我们提出了新的算法计算伴随理想的曲线，因此，在平面的情况下，伴随曲线。关于术语，我们遵循Gorenstein，他用导体来陈述伴随条件。我们的主要算法产生的Gorenstein伴随理想G的一个给定的曲线，我们称之为当地Gorenstein伴随理想的交集。由于各自的本地计算不依赖于彼此，我们的方法本质上是并行的。在有理数上，进一步的并行化通过算法的模块化版本来实现，该算法首先计算G的特征p对应物的数量，然后将这些提升到特征零。作为一个关键的组成部分，我们建立了一个有效的标准来验证升力的正确性。著名的应用是Riemann-Roch空间的计算，模空间中点的构造，以及有理曲线的参数化。我们已经实现了不同的变种，我们的算法连同Mnuk的方法在计算机代数系统奇异，并给出时间来比较算法的性能。

The number of equisingular moduli of a rational surface singularity

We consider a conjectured topological inequality for the number of equisingular moduli of a rational surface singularity, and prove it in some natural special cases. When the resolution dual graph is "sufficiently negative" (in a precise sense), we verify the inequality via an easy cohomological vanishing theorem, which implies that this number is computed simply from the graph (Theorem 3.10). To consider an important and less restrictive meaning of "sufficiently negative" requires a much more difficult "hard vanishing theorem" (Theorem 4.5), which is false in characteristic p. Theorem 7.9 ver…

关于有理复曲面上相对极小纤维化的斜率

On the slope of relatively minimal fibrations on rational complex surfaces

给定有理曲面$S$上的一个相对极小纤维化$f: S \to \Bbb P^1$，其一般纤维$C$为$g$亏格，研究了不等式$6(g-1)\le K_f^2$在什么条件下成立，其中$K_f$是$f$的典型相对层.我们根据$C$的亏格和角性以及$S$上某些例外曲线的个数，给出了不等式成立的充分条件.我们说明了如何将这些结果可用于构建所需的属性的纤维化。对于亏格为$11\le g\le 49$的纤维化，我们证明了如下不等式：$$ 6(g-1) +4 -4\sqrt g \le K_f^2.$$

Line and rational curve arrangements, and Walther's inequality

There are two invariants associated to any line arrangement: the freeness defect $ν(C)$ and an upper bound for it, denoted by $ν'(C)$, coming from a recent result by Uli Walther. We show that $ν'(C)$ is combinatorially determined, at least when the number of lines in $C$ is odd, while the same property is conjectural for $ν(C)$. In addition, we conjecture that the equality $ν(C)=ν'(C)$ holds if and only if the essential arrangement $C$ of $d$ lines has either a point of multiplicity $d-1$, or has only double and triple points. We prove both conjectures in some cases, in particular when the num…

奇妙群紧化上的极小有理曲线

Minimal rational curves on wonderful group compactifications

考虑伴随型单代数群G及其奇妙的紧化X.证明了X存在唯一的极小有理曲线族，并明确地描述了通过一般点的曲线族。作为应用，我们证明了当G不是A_1型或C型时，X具有目标刚性性质。

有理映射因子系数的最大比与高度的上界

Maximal ratio of coefficients of divisors and an upper bound for height for rational maps

当我们有一个态射f：P^n -> P^n时，我们有一个不等式\frac{1}{°f} h（f（P））+C > h（P），它提供了$h(P)$的一个好的上界。然而，如果$f$是有理映射，则\frac{1}{°f} h（f（P））+C不能是h（P）的上界。本文定义了有理映射$f$的$D$ -比，用它来代替h（P）的高度不等式中态射的度.

On the projective normality of cyclic coverings over a rational surface

Let $S$ be a rational surface with $\dim|-K_S|\ge 1$ and let $π: X\rightarrow S$ be a ramified cyclic covering from a nonruled smooth surface $X$. We show that for any integer $k\ge 3$ and ample divisor $A$ on $S$, the adjoint divisor $K_X+kπ^*A$ is very ample and normally generated. Similar result holds for minimal (possibly singular) coverings.

至少含6根奇异纤维的有理曲面上的半稳定纤维化

Semistable Fibrations on Rational Surfaces with at least 6 Singular Fibers

这篇论文被撤回了。一个完全重写的新版本将很快提交。为了得到不等式$6(g-1) \le K_f^2$，必须对纤维化施加一些附加条件，包括某些例外曲线的数目。

On the projective normality of double coverings over a rational surface

We study the projective normality of a minimal surface $X$ which is a ramified double covering over a rational surface $S$ with $\dim|-K_S|\ge 1$. In particular Horikawa surfaces, the minimal surfaces of general type with $K^2_X=2p_g(X)-4$, are of this type, up to resolution of singularities. Let $π$ be the covering map from $X$ to $S$. We show that the $\mathbb{Z}_2$-invariant adjoint divisors $K_X+rπ^*A$ are normally generated, where the integer $r\ge 3$ and $A$ is an ample divisor on $S$.

Delta invariant of curves on rational surfaces I. The analytic approach

We prove that if (C,0) is a reduced curve germ on a rational surface singularity (X,0) then its delta invariant can be recovered by a concrete expression associated with the embedded topological type of the pair (X,C). Furthermore, we also identify it with another (a priori) embedded analytic invariant, which is motivated by the theory of adjoint ideals. Finally, we connect our formulae with the local correction term at singular points of the global Riemann--Roch formula, valid for projective normal surfaces, introduced by Blache.

组合Li-Yau不等式与曲线上的有理点

A combinatorial Li-Yau inequality and rational points on curves

提出了一种基于图论的非阿基米德曲线的角度控制方法。设k表示完备非阿基米德值域。我们首先证明了一个下界的一条曲线的代数闭包的k在一类图映射的最小程度，即：一个应该最小化所有所谓的有限调和图态射树，起源于任何细化的对偶图的稳定模型的曲线。接下来是我们的主要结果：我们证明了一个下界的程度，这样的一个图态射的拉普拉斯算子的第一个特征值和一些“体积”的原始图，这可以被看作是一个替代图的李-丘不等式从微分几何，虽然我们也证明了严格的模拟原始不等式失败的一般图。最后，我们应用这些结果给出了有限域上任意Drinfeld模曲线的一致性的下界，以及Gamma（1）的一般同余子群Gamma在指数[Gamma（1）：Gamma]上是线性的，其常数仅取决于剩余域的度和所选“无限”位置的度.这是经典模曲线的阿布拉莫维奇定理的函数域模拟。我们目前的应用程序的一致有界性的扭转秩两Drinfeld模块，改善现有的结果，并在一定的椭圆曲线的模块化程度的下界的功能领域，解决了问题的Papikian。

A degree bound for families of rational curves on surfaces

We give an upper bound for the degree of rational curves in a family that covers a given birational ruled surface in projective space. The upper bound is stated in terms of the degree, sectional genus and arithmetic genus of the surface. We introduce an algorithm for constructing examples where the upper bound is tight. As an application of our methods we improve an inequality on lattice polygons.

On the Bogomolov-Gieseker inequality for tame Deligne-Mumford surfaces

We generalize the Bogomolov-Gieseker inequality for semistable coherent sheaves on smooth projective surfaces to smooth Deligne-Mumford surfaces. We work over positive characteristic $p>0$ and generalize Langer's method to smooth Deligne-Mumford stacks. As applications we obtain the Bogomolov inequality for semistable coherent sheaves on a Deligne-Mumford surface in characteristic zero, and the Bogomolov inequality for semistable sheaves on a root stack over a smooth surface which is equivalent to the Bogomolov inequality for the rational parabolic sheaves on a smooth surface $S$. In a joint a…