Dirichlet fundamental domains and complex-projective varieties
Michael Kapovich
证明了对每个有限表示群G，存在一个以基本群G为中心的2维不可约复投射簇W，使得W的所有奇点都是正规交叉和Whitney伞.
We prove that for every finitely-presented group G there exists a 2-dimensional irreducible complex-projective variety W with the fundamental group G, so that all singularities of W are normal crossings and Whitney umbrellas.
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On classification of projective homogeneous varieties up to motivic isomorphism
本文给出了维数小于小于6的各向异性投射齐次簇到motivic同构的一个完全分类。本文给出了A_n型各向异性旗簇具有同构基模的几个判别准则。
Chern numbers and diffeomorphism types of projective varieties
在1954年Hirzebruch问线性组合的陈数是拓扑不变量顺利复杂的投影品种。我们给出了一个完整的答案，这个问题在小尺寸，也证明了部分结果没有限制的尺寸。
We determine which complex abelian varieties can be realized as the automorphism group of a smooth projective variety.
Euler-symmetric projective varieties are nondegenerate projective varieties admitting many C*-actions of Euler type. They are quasi-homogeneous and uniquely determined by their fundamental forms at a general point. We show that Euler-symmetric projective varieties can be classified by symbol systems, a class of algebraic objects modeled on the systems of fundamental forms at general points of projective varieties. We study relations between the algebraic properties of symbol systems and the geometric properties of Euler-symmetric projective varieties. We describe also the relation between Eule…
We show that smooth projective horospherical varieties with nef tangent bundles are rational homogeneous spaces.
Algebraic dynamical systems and Dirichlet's unit theorem on arithmetic varieties
本文从两个方面研究了Dirichlet性质的障碍：非正点的稠密性和广义R-因子上的泛函。应用于代数动力系统，这些结果提供了不具有Dirichlet性质的Nef代数算术R-Cartier因子的例子.我们希望文中所得到的障碍能为狄利克雷性质的判定提供一条道路。
In this paper we show that quotients of smooth projective toric varieties by $μ_p$ in positive characteristics $p$ are toric varieties.
On projective varieties with nef anticanonical divisors
证明了具有nef反正则因子的投射簇的一个结构定理。
We show an analogue of Jordan's theorem for algebraic groups defined over a field $\mathbb k$ of arbitrary characteristic. As a consequence, a Jordan-type property holds for the automorphism group of any projective variety over $\mathbb k$.
Constructing projective varieties in weighted flag varieties
本文计算了广义加权旗簇的Hilbert级数，并讨论了确定其定义方程的计算机辅助方法。我们将所得结果应用于G_2型和GL（6）型李群的加权旗簇，分别构造了余维为8和6的极化投射簇.
Classification of embedded projective manifolds swept out by rational homogeneous varieties of codimension one
给出了一类被Picard数和余维数均为1的有理齐次簇扫出的嵌入光滑射影簇的分类。
Toric subvarieties of projective space are classified up to projective automorphisms.
Let $A$ be an abelian variety over an algebraically closed field. We show that $A$ is the automorphism group scheme of some smooth projective variety if and only if $A$ has only finitely many automorphisms as an algebraic group. This generalizes a result of Lombardo and Maffei for complex abelian varieties.
It is known that projective minimal models satisfy the celebrated Miyaoka-Yau inequalities. In this article, we extend these inequalities to the set of all smooth, projective and non-uniruled varieties.
We give a brief systematic overview of a few results concerning the Néron--Severi lattices of Fermat varieties and Delsarte surfaces.
Divisors on Projective Hibi Varieties
以Hibi环为齐次坐标环，计算了射影簇的除子类群和Picard群.这些变种正是与有序多面体相关的环面变种。我们使用的工具，从理论的环面簇得到的描述，这两个群体只依赖于组合性质的基础偏序集。
On the osculatory behaviour of higher dimensional projective varieties
在这里，我们探讨的密切空间的几何任意维的射影品种。特别是，我们分类品种非常退化的高阶密切空间，我们确定温和的条件存在的拐点。
Symmetric varieties are normal equivariant open embeddings of symmetric homogeneous spaces and they are interesting examples of spherical varieties. The principal goal of this article is to study the rigidity under Kähler deformations of smooth projective symmetric varieties with Picard number one.
The Lefschetz algebra $L^*(X)$ of a smooth complex projective variety $X$ is the subalgebra of the cohomology algebra of $X$ generated by divisor classes. We construct smooth complex projective varieties whose Lefschetz algebras do not satisfy analogues of the hard Lefschetz theorem and Poincaré duality.
Quasi--Projective Reduction of Toric Varieties
定义复代数簇$X$的拟投射约化为从$X$到拟投射簇的正则映射，该映射关于从$X$到拟射影簇的正则映射是泛映射.环面拟投射约化是环面簇范畴中的类似概念。对于给定的复曲面簇$X$，我们首先构造一个复曲面拟投射约化。然后证明了$X$有拟投射约化当且仅当它的环面拟投射约化是满射的。我们应用这个结果来刻画当一个子环面作用在一个拟投射环面簇上时，它在拟射影簇范畴中容许一个范畴商。
In this note we introduce the concept of reflective projective varieties. These are stratified projective varieties with certain dimension constraints on their dual varieties. We prove that for such varieties, the Chern-Schwartz-MacPherson classes of the strata completely determine the local Euler obstructions and the polar degrees. We also propose an algorithm to compute the local Euler obstructions when such varieties are formed by group orbits. As examples we compute the local Euler obstructions of quadratic hypersurfaces and ordinary determinantal varieties to illustrate our method.
We study the equivariant cobordism rings for the action of a torus $T$ on smooth varieties over an algebraically closed field of characteristic zero. We prove a theorem describing the rational $T$-equivariant cobordism rings of smooth projective $G$-spherical varieties with the action of a maximal torus $T$ of $G$. As an application, we obtain explicit presentations for the rational equivariant cobordism rings of smooth projective horospherical varieties of Picard number one.
Rigid geometry on projective varieties
我们证明了各种类型的全纯抛物几何光滑复射影簇的刚性。
Smooth projective horospherical varieties with Picard number 1
我们描述光滑的投影horosphere品种皮卡数为1。此外，我们证明了任何这样的簇的自同构群的作用至多有两个轨道，我们给出了非齐次的几何特征。
We construct irreducible complex projective varieties such that the Lyubeznik numbers of their affine cones depend on the choices of projective embeddings. The main ingredient is the recent work of Reichelt-Saito-Walther, where the Lyubeznik numbers are reinterpreted using perverse sheaves and reducible projective varieties with dependent Lyubeznik numbers are constructed.
A Remark on Projective Embeddings of Varieties with Non-Negative Cotangent Bundles
我们证明了一个关于光滑簇X的投影嵌入的基本的但有点出乎意料的结果，其余切丛是数值有效的。具体地说，我们证明了任何投射嵌入中X的度必须在X的维度上（几乎）指数增长。
We show that the moduli space of rank three toric vector bundles on smooth projective toric varieties satisfies Murphy's law, that is, every singularity type of finite type arises somewhere on this space.
Toward Dirichlet's unit theorem on arithmetic varieties
在本文中，我们想提出一个基本问题，高维类似的狄利克雷单位定理。作为算术Hodge指数定理的一个应用，我们也给出了这个问题的部分答案。
The geometric and algebraic properties of smooth projective varieties with 1-regular structure sheaf are well understood, and the complete classification of these varieties is a classical result. The aim of this paper is to study the next case: smooth projective varieties with 2-regular structure sheaf. First, we give a classification of such varieties using adjunction mappings. Next, under suitable conditions, we study the syzygies of section rings of those varieties to understand the structure of the Betti tables, and show a sharp bound for Castelnuovo-Mumford regularity.
Computing Segre classes in arbitrary projective varieties
通过计算线性投影序列的度，给出了计算任意投影簇的子模式的Segre类的一个算法。基于Segre类的射影簇通过一般有效Cartier因子与交可交换的事实，我们可以编制一个线性方程组，它决定了将Segre类推进到射影空间的系数.这里提出的算法来后，其他几个解决问题的特殊情况下，其中的环境品种是例如射影空间;据我们所知，这是第一个算法能够计算塞格雷类的投影品种与任意奇点。
Linear and Steiner bundles on projective varieties
利用算术Cohen-Macaulay（ACM）簇的Horrocks单子的推广，建立了射影空间和二次超曲面上线性丛和Steiner丛的上同调刻画.我们还研究了ACM簇上丛的线丛分解，并在二次超曲面的情况下刻画了线性同调维数。
On some smooth projective two-orbits varieties with Picard number 1
我们分类所有光滑的投影horosphere品种皮卡德数1。我们证明了任何这样的簇X的自同构群最多有两个轨道作用，并且这个群仍然只有两个轨道作用在X上的闭轨道上。我们的特点是所有光滑的投影两轨道品种皮卡德数1，满足后一个属性。
Quadratic equations of projective $PGL_2(\C)$-varieties
本文给出了由二次曲面定义的任意射影$PGL_2(\C)$ -簇的显式方程。我们研究了它们的零轨迹和它们与Veronese曲线几何的关系。
We shortly describe the algorithms behind some of the functions provided by the Macaulay2 package MultiprojectiveVarieties, a package for multi-projective varieties and rational maps between them.
We classify the Ulrich vector bundles of arbitrary rank on smooth projective varieties of minimal degree. In the process, we prove the stability of the sheaves of relative differentials on rational scrolls.
Identifying quadric bundle structures on complex projective varieties
本文利用某些有理曲线族的几何性质，刻画了在稠密开子集上具有二次丛结构的光滑复射影簇。
Ample bodies and Terracini loci of projective varieties
本文引入射影簇的充足体概念，用以证明 Terracini 轨迹为空，并给出环面簇与齐次簇的特定可辨识性结论。
Smooth projective toric varieties whose nontrivial nef line bundles are big
对于任意的$n\geq 3$，我们明确地构造了Picard数$\geq 5$的光滑投射环面$n$ -折叠，其中任意非平凡的Nef线丛都是大的.
Let $G$ be a semi-simple algebraic group over a perfect field $k$. A lot of progress has been made recently in computing the Chow motives of projective $G$-homogenous varieties. When $k$ has positive characteristic, a broader class of $G$-homogeneous varieties appear. These are varieties over which $G$ acts transitively with possibly non-reduced isotropy subgroup. In this paper we study these varieties which we call ${\it \mbox{projective pseudo-homogeneous varieties}}$ for $G$ inner type over $k$ and prove that their motives satisfy Rost nilpotence. We also find their motivic decompositions a…
Some results about the projective normality of abelian varieties
我们把极化阿贝尔簇的投影正规性问题归结为检查非常显式矩阵的秩。这使得我们可以证明关于三重和四重交换上本原线丛正规生成的一些结果。我们还给出了两种情况下，投影正规性总是失败。最后，我们做了一些猜测。
Our main aim is to provide a uniform geometric characterization of the analogues over arbitrary fields of the four complex Severi varieties, i.e.~the quadric Veronese varieties in 5-dimensional projective spaces, the Segre varieties in 8-di\-men\-sional projective spaces, the line Grassmannians in 14-dimensional projective spaces, and the exceptional varieties of type $\mathsf{E}_{6}$ in 26-dimensional projective space. Our theorem can be regarded as a far-reaching generalization of Mazzocca and Melone's approach to finite quadric Veronesean varieties. This approach takes projective properties…
Stability of holonomicity over quasi-projective varieties
设$\V$是具有完美剩余域$k$的混合特征完备离散赋值环.我们解决了Berthelot关于光滑射影形式$\V$ -格式上完整性稳定性的问题。然后我们建立了一类具有有界、$F$ -完整上同调的拟投射$k$ -簇上的算术$\D$ -模的复形.在Grothendieck的六种操作下，我们得到了它的稳定性。
Projective normality of abelian varieties
我们证明了g维单阿贝尔簇上的充足线丛L，其中h^0（L）> 2^g.g！，给出射影正规嵌入，对于所有g>0。
Automorphisms of Certain Projective Bundles over Toric Varieties
当丛E是线丛的直和时，刻画了单复曲面簇上投射丛P（E）的自同构群。复曲面品种，包括卡拉比-丘m-褶皱和Del Pezzo表面上的完整的交叉点的模量的研究中的应用，表明。
Cohomological rank functions and syzygies of projective bundles on abelian varieties
本文利用Jiang–Pareschi上同调秩函数及Caucci–Ito技术研究阿贝尔簇上射影丛的合冲，从而推广并细化了Chiantipalli关于性质$N_p$的结果。
Galois subspaces for projective varieties
该研究针对射影簇到射影空间的嵌入场景，探究满足“与嵌入复合后形成同维射影空间间Galois态射”的线性投影空间的结构，属于代数几何领域的理论研究。
Adelic line bundles on quasi-projective varieties
本书构建有限生成域上拟射影簇的阿代尔线丛理论，探讨其相交、体积与高度理论，并应用于拟射影簇代数点的高度研究。
Algebraic varieties with quasi-projective universal cover
本文证明了正规投射簇X的泛覆盖是拟投射的当且仅当X的有限等距覆盖是具有单连通纤维的阿贝尔簇上的纤维丛。
We define partially ample subvarieties of projective varieties, generalizing Ottem's work on ample subvarieties, and show their ubiquity. As an application, we obtain a connectedness result for pre-images of subvarieties by morphisms, reminiscent to a problem posed by Fulton-Hansen.
We give an example of a pair of projective symplectic varieties in arbitrarily large dimensions which are D-equivalent, L-equivalent, and birationally inequivalent.
Segre Classes on Smooth Projective Toric Varieties
本文推广了Eklund-Jost-Peterson的算法，用于计算射影k-空间的闭子概形的Segre类.本文将该算法推广到计算光滑投射环面簇的闭子概形的Segre类。
A Derived Equivalence for some Twisted Projective Homogeneous Varieties
本文构造了Severi-Brauer簇和对合簇的倾斜层。这个层将每个簇的导出范畴与环上模的导出范畴联系起来，环的半单分量由相应线性代数群的Tits代数组成。
In this article we prove new results on projective normality and normal presentation of adjunction bundle associated to an ample and globally generated line bundle on higher dimensional smooth projective varieties with nef canonical bundle. As one of the consequences of the main theorem, we give bounds on very ampleness and projective normality of pluricanonical linear systems on varieties of general type in dimensions three, four and five. These improve known such results.
We prove a theorem of Tits type for automorphism groups of projective varieties over an algebraically closed field of arbitrary characteristic, which was first conjectured by Keum, Oguiso and Zhang for complex projective varieties.
Characterizing Projective Spaces for Varieties with at Most Quotient Singularities
本文将Cho，Miyaoka和Shepherd-Barron提出的射影空间的数值判据推广到最坏商奇点的簇.设$X$是维数为$n \geq 3$的正规射影簇，至多具有商奇点.我们的结果断言，如果$C \cdot (-K_X) \geq n+1$对每个曲线$C \subset X$，那么$X \cong \PP^n$。
We give evidence for a uniformization-type conjecture, that any algebraic variety can be altered into a variety endowed with a tower of smooth fibrations of relative dimension one.
We study the projective behavior, mainly with respect to osculating spaces and secant varieties, of Lagrangian Grassmannians and Spinor varieties. We prove that these varieties have osculating dimension smaller than expected. Furthermore, we give numerical conditions ensuring the non secant defectivity of Lagrangian Grassmannians in their Plücker embedding and of Spinor varieties in both their Plücker and Spinor embeddings.
Normal projective varieties of degree 5
我们列出了不可约的减少和不退化的正常的投影品种$X\subset\mathbb{P}^N$的尺寸$n$和度5定义在一个代数闭域$k$的字符$(k) = 0$。在光滑的情况下，或者当$n = 2$时，我们还给出char $(k) \geq 0$的任何代数闭域$k$的分类结果。
Nilpotent groups and universal coverings of smooth projective varieties
本文证明了具有幂零基本群的光滑投射簇的泛覆盖是全纯凸的。
Quantum cohomology of smooth complete intersections in weighted projective spaces and singular toric varieties
我们推广Givental定理完全相交光滑环面品种在Fano的情况。特别是，我们发现Gromov-Witten不变量的Fano品种的尺寸$\geq 3$，这是完全的交叉加权射影空间和奇异环面品种。我们将Riemann-Roch方程推广到加权射影空间。我们计算计数矩阵的光滑Fano三倍皮卡德群Z和anticanonical度2，8，和16。
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