John Kopper
Using Bridgeland stability conditions we give sufficient criteria for a stable vector bundle on a surface to remain stable when restricted to a curve. We give a stronger criterion when the vector bundle is a general vector bundle on the plane. As an application, we compute the cohomology of such bundles for curves that lie in the plane or on Hirzebruch surfaces.
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Isomorphisms and automorphisms of multiprojective bundles and symmetric powers of projective bundles
本文研究射影空间上多重射影丛及射影丛相对对称幂的同构分类问题，并进一步刻画了这些对象的自同构群。
In this short article, we determine the bigness of the tangent bundle $T_X$ of the projective bundle $X=\mathbb{P}_C(E)$ associated to a vector bundle $E$ on a smooth projective curve $C$.
We study global deformations of certain projective bundles over projective spaces. We show that any projective global deformation of a projective bundle over $\bP^1$ carries the structure of a projective bundle over some projective space. Furthermore, we construct examples in arbitrary dimension $\ge 3$ of almost homogeneous Fano projective bundles over $\bP^2$ which can be globally deformed to non-almost homogeneous manifolds.
Complements of hyperplane sub-bundles in projective space bundles over the projective line
我们证明了在射影直线上秩为r-1的射影空间丛的充要超平面子丛H的补的抽象代数簇的同构型只依赖于H的r重自交.特别地，它既不依赖于环境丛，也不依赖于具有给定r重自相交的特定的充足超平面子丛。我们的证明利用了意想不到的财产，每一个这样的补充配备了一个非平凡torsor下的向量丛的仿射直线上的双重起源的结构。
The aim of this note is to investigate the relation between two types of non-singular projective varieties of Picard rank 2, namely the Projective bundles over Projective spaces and certain Blow-up of Projective spaces.
Isomorphism of Multiprojective Bundles and Projective Towers
本文研究不同维数光滑射影簇上射影丛的同构问题，证明不同射影空间上的多射影丛除平凡情形外不同构，并给出高度3射影塔顶层簇同构的充要条件。
We show that smooth projective horospherical varieties with nef tangent bundles are rational homogeneous spaces.
Dynamics of Endomorphisms for Projective Bundles on Elliptic Curves
本文通过丛的转移函数研究椭圆曲线上射影丛满射自同态的动力学，证明了其Kawaguchi--Silverman猜想，并将结果推广至具有限生成Mori锥的光滑射影簇上的射影分裂丛。
On the classification of rank 2 almost Fano bundles on projective space
几乎Fano丛是光滑射影簇上的向量丛，它的射影化是几乎Fano簇。本文证明了几乎Fano丛只存在于几乎Fano流形上，并研究了射影空间上的秩2几乎Fano丛。
In previous papers it was shown that the left and right O-module structure of the jet bundles on the projective line differed. In this paper we show that similar statements hold for jet bundles on projective space in any dimension. We also consider some classes associated to jet bundles in the algebraic K-theory of projective space.
Smooth blow up structures on projective bundles
本文在Hartshorne猜想等假设下分类了射影空间上具有光滑爆破结构的射影丛，验证了特定向量丛的相关性质，并得到了同时具有两种结构的簇的新例子。
We study ample vector bundles on smooth projective stacks. In particular, we prove that the tangent bundle for the weighted projective stack $\mathbb{P}(a_0,...,a_n)$ is ample. A result of Mori shows that the only smooth projective varieties with ample vector bundle are isomorphic to $\mathbb{P}^N$ for some $N$. Extending our geometric spaces from varieties to projective stacks, we are able to provide a new example. This leaves the open question of knowing if the only smooth projective stacks are the weighted projective stacks.
This short paper shows that a projective bundle over a smooth variety with a section can be uniquely recovered from the embedding of the section in its second infinitesimal neighborhood within the bundle.
On Serrano's conjecture on Projective bundles
本文针对高维光滑射影簇上的射影丛，考察严格 nef 除子情形下的 Serrano 猜想。
On projective space bundle with nef normalized tautological divisor
本文研究了相对反标准线丛为nef的射影空间丛的结构。作为应用，我们得到了有限余覆盖下的阿贝尔簇的一个刻画。
Frobenius splitting of projective toric bundles
给出了非奇异环面簇上的环面丛的几个温和条件，在这些条件下环面丛的射影化是Frobenius分裂的。
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。
Schwarzenberger bundles on smooth projective varieties
定义了任意光滑射影簇X上的Schwarzenberger丛。本文引入了Steiner丛E在X上的跳跃对的概念，并确定了其跳跃轨迹维数的一个界。对跳对集具有最大维数的Steiner丛进行了完全分类，证明了它们都是Schwarzenberger丛。
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.
On globally generated vector bundles on projective spaces II
推广了作者以前的一个结果，对第一陈类为3的射影空间上的整体生成向量丛进行了分类。
Given two equivariant vector bundles over an algebraic GKM manifold with the same equivariant Chern classes, we show that the genus zero equivariant Gromov--Witten theory of their projective bundles are naturally isomorphic.
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。
Restrictions of stable bundles
Mehta-Ramanathan定理保证了一个稳定的向量丛对一个足够高次的完全相交曲线的限制也是稳定的。我们提高了“足够高的程度”的界限，并提出了一个可能的最佳猜想。
Projective manifolds with splitting tangent bundles, I
研究了Kodaira维数不为1和2时，其切丛是子丛的直和的射影三重图的泛覆盖.我们还证明了在任意维上具有分裂切丛的Fano流形的结果。
We systematically study the splitting of vector bundles on a smooth, projective variety, whose restriction to the zero locus of a regular section of an ample vector bundle splits. First, we find ampleness and genericity conditions which ensure that the splitting of the vector bundle along the subvariety implies its global splitting. Second, we obtain a simple splitting criterion for vector bundles on the Grassmannian and on partial flag varieties.
Hartshorne's conjecture about vector bundles on projective space states that any rank 2 vector bundle on n-dimensional projective space splits as soon as n is at least 7. Klyachko has shown that Hartshorne's conjecture is true when the vector bundles are torus equivariant. Moreover, recent work of Ilten and Süss generalizes Klyachko's work to the case of a smaller rank torus action on projective space. In this note we give a new, direct proof that torus rank 2 bundles split that avoids a description of the category of torus equivariant vector bundles.
We classify singular fibres of a projective Lagrangian fibration over codimension one points. As an application, we obtain a canonical bundle formula for a projective Lagrangian fibration over a smooth manifold.
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。
Our goal is to study the syzygies of the projective embeddings defined by ample line bundles on a fake projective plane S. The syzygies are studied in terms of the property $N_p$. For various kinds of ample line bundles, we give explicit lower bounds for their powers above which the property $N_p$ is satisfied.
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.
Restriction of stable bundles on a jacobian of genus 2 to an embedded curve
本文的目的是描述从2维Jacobian $X$上具有小$c_2$的稳定秩2丛的模空间到嵌入$X$中的亏格2曲线$C$上的稳定秩2丛的模空间的限制映射.
Matrices of linear forms of constant rank from vector bundles on projective spaces
本文利用射影空间上小第一陈类的全局生成向量丛分类，完全刻画了秩不超过六的不可分解常秩线性形式矩阵，并揭示了其与Steiner丛、Drézet丛、刚性齐次向量丛及instanton丛的联系。
The cones of effective cycles on projective bundles over curves
推广Miyaoka等人在除子和曲线情形下的工作，利用Harder-Narasimhan滤子中的数值数据计算复射影曲线上射影丛上任意维数有效圈的锥.最后给出了任意维光滑复射影基上射影丛上的圈的一个应用。
Projective bundles of singular plane cubics
分类理论和研究由最小次数的有理曲线所覆盖的射影簇自然会导致奇异有理曲线族的研究。由于家庭的任意奇异曲线是很难处理的，它已被证明在以前的文件中，存在一个部分的奇点的分解，它转换成一束可能严重奇异曲线束节点和尖点平面三次。在分类论感兴趣的情况下，这些丛的全空间显然是投射的。然而，平面三次体的任意丛是全局射影的一般是错误的。由于这个原因，投射性的问题似乎是有趣的，目前的工作给出了一个特征的投射丛。
Convex Fujita numbers: projective bundles
本文研究曲线及阿贝尔簇上曲线半稳定向量丛射影化的有效整体生成性质，属于代数几何领域凸Fujita数相关研究。
On the projective normality of Ulrich bundles on some low-dimensional varieties
本文研究了 Ulrich 向量丛通过其 tautological 线丛的完全线性系嵌入时的射影丛的射影正规性，重点关注定义于曲线、$q=p_g=0$ 曲面及 $2$ 和 $3$ 维超曲面上的 Ulrich 丛。
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.
Characterization theorems for the projective space and vector bundle adjunction
我们考虑光滑射影簇X实际上是射影空间的一些条件。我们还将向量丛的一些结果推广到正特征的情形。我们使用的方法和技术的所谓森理论，特别是研究有理曲线的射影流形。
Quantum cohomology of projective bundles over P^n
本文试图确定射影空间P^n上射影丛的量子上同调。与前面的例子相反，在我们的例子中，相关的模空间通常没有期望的维数。这使得计算更加困难。我们通过使用过度相交理论克服了这个困难。
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.
A Babylonian tower theorem for principal bundles over projective spaces
我们推广了由I。Coanda和G. Trautmann（2006）的方法应用于射影空间上的主$G$ -丛的情形，其中$G$是定义在代数闭域上的线性代数群。在证明的过程中，对这种主$G$ -丛的结构有了新的认识。
Normal bundles of rational curves in projective spaces
我们确定的分裂（同构）类型的一般属0曲线在任何射影空间的1或2个组件的正常丛，以及（有时非平凡的）的方式束变形局部与一般变形的曲线。本文导出了法丛为非一般分裂型的光滑有理曲线除迹线的计数公式。
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。
Infinitely stably extendable vector bundles on projective spaces
根据Horrocks（1966），射影n-空间上的一个向量丛E稳定地扩张到射影N-空间，N>n，如果在较大的空间上存在一个向量丛，它对较小的向量丛的限制同构于E加上一个线丛的直和。我们证明了对每个N>n，E稳定地扩张到投射N-空间当且仅当E是自由单子（有三项）的上同调。证明使用Coanda和Trautmann（2006）的方法。结合Mohan Kumar，Peterson和Rao（2003）的一个定理，我们得到了射影空间上向量丛的巴比伦塔定理的一个新的有效形式.
Let $U^{'s}_L(n,d)$ be the moduli space of stable vector bundles of rank $n$ with determinant $L$ where $L$ is a fixed line bundle of degree $d$ over a nodal curve $Y$. We prove that the projective Poincare bundle on $Y \times U^{'s}_L(n,d)$ and the projective Picard bundle on $U^{'s}_L(n,d)$ are stable for suitable polarisation. For a nonsingular point $x \in Y$, we show that the restriction of the projective Poincare bundle to $x \times U^{'s}_L(n,d)$ is stable for any polarisation. We prove that for arithmetic genus $g\ge 3$ and for $g=n=2, d$ odd, the Picard group of the moduli space $U'_L…
P1-bundles over projective manifolds of Picard number one each of which admit another smooth morphism of relative dimension one
本文给出了Picard数为1的射影流形上的P1-丛的一个完全分类，其中P1-丛允许另一个相对维数为1的光滑态射。
A stratified bundle is a vector bundle which is a D-module. We show that regular singularity of stratified bundles on smooth varieties in positive characteristic is preserved by pullback and that regular singularity can be checked on curves, if the ground field is large enough.
We prove that the kernel bundle of the evaluation morphism of global sections, namely the syzygy bundle, of a sufficiently ample line bundle on a smooth projective variety is slope stable with respect to any polarization. This settles a conjecture of Ein-Lazarsfeld-Mustopa.
In this article, we study Newton-Okounkov bodies on projective vector bundles over curves. Inspired by Wolfe's estimates used to compute the volume function on these varieties, we compute all Newton-Okounkov bodies with respect to linear flags. Moreover, we characterize semi-stable vector bundles over curves via Newton-Okounkov bodies.
On Ulrich bundles on projective bundles
本文研究射影丛上Ulrich向量丛的存在性，建立了底簇与射影丛上Ulrich向量丛的关联，并由此在曲线和曲面上的射影丛上证明了其存在性。
Cohomological rank functions and syzygies of projective bundles on abelian varieties
本文利用Jiang–Pareschi上同调秩函数及Caucci–Ito技术研究阿贝尔簇上射影丛的合冲，从而推广并细化了Chiantipalli关于性质$N_p$的结果。
In this note we give a complete description of all the hyperplane section of the projective bundle associated to the tangent bundle of $\mathbb{P}^2$ under its natural embedding in $\mathbb{P}^7.$ As an application one obtains a description of all possible deformations, in $\text{SL_3(\mathbb{C}}/B,$ of the co-dimension one sub-scheme which is the union of two fundamental Schubert divisors.
Let E be a stable rank two normalized vector bundle on P3. If H is a general plane we show that h^0(E_H(1)) \leq 2+c_1. It follows that h^0(E(1)) \leq 2+c_1.
Mirror symmetry and quantum cohomology of projective bundles
在以前的文章中，我们证明了光滑簇$X$的量子$\mathcal D$ -模与其上的线丛直和的投影之间的关系，本文证明了当$X$是环面簇中的完全交时的猜想。我们还利用这个猜想证明了由微分算子生成的小量子上同调环$X$的关系提升到投射丛。射影丛的基本上同调关系变形为小量子上同调关系。
Let $X$ be a projective manifold of dimension $n$. Suppose that $T_X$ contains an ample subsheaf. We show that $X$ is isomorphic to $\mathbb{P}^n$. As an application, we derive the classification of projective manifolds containing a $\mathbb{P}^r$-bundle as an ample divisor by the recent work of D.~Litt.
p-Buchsbaum rank 2 bundles on the projective space
许多作者已经证明了P3上的归一化的1-Buchsbaum秩2向量丛是零相关丛，而P3上的归一化的2-Buchsbaum秩2向量丛是电荷为2的瞬子丛。我们发现P3上的3-Buchsbaum秩为2的向量丛不具有同样的性质，并提出了一个关于此类对象分类的猜想.
On the quantum cohomology of Fano bundles over projective spaces
在他们的论文“Quantum cohomology of projective bundles over $P^n$“（Trans. Am. Math. Soc.（1998）350：9 3615-3638）Z.Qin和Y.Ruan介绍了计算流形的量子环的有趣技巧，这些流形是射影空间上的射影化丛;特别地，在分裂丛的情况下，他们在一些限制下证明了关于环面流形的量子环的Batyrev公式。在这里，我们证明了Batyrev公式的量子Chern-Leray方程的一大类分裂丛。
A mirror conjecture for projective bundles
提出并证明了光滑复射影流形$X$和$X$上的射影丛$\PP(\oplus L_i)$的量子上同调的$\mathcal D$ -模之间的关系.
A Torelli type problem for logarithmic bundles over projective varieties
研究了光滑射影簇中固定次数超曲面的对数丛。然后，我们专门的情况下，品种是一个二次超曲面和多投射空间证明Torelli型定理在某些情况下。
Ulrich bundles on double covers of projective spaces
本文证明了任意射影空间二重覆盖的光滑射影簇都容许Ulrich丛，并在二维情形下构造了秩为二的Ulrich丛。
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