Victor Pretti
In this paper we examine the conditions that an object $E$ with $\text{ch}_0(E)=0$ has to satisfy in order for it to be asymptotically (semi)stable with regard to Weak or Bridgeland stability conditions. This notion turned out to be equivalent to sheaf Gieseker-Simpson (semi)stability or a dual of it, depending on the curve considered.
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Bridgeland Stability of Sheaves on del Pezzo Surface of Picard Rank Three
该研究针对皮卡秩3的德尔佩佐曲面（两点吹胀的$\u2119^{2}$）上的层，确定了线丛的失稳对象，证明了任意$(-1)$-曲线的限制层在所有除子型Bridgeland稳定性条件下均稳定，推进了代数几何中Bridgeland稳定性的具体计算研究。
Rank One Bridgeland Stable Moduli Spaces on A Principally Polarized Abelian Surface
我们计算了Bridgeland稳定对象在不可约主极化复阿贝尔曲面上对应于扭曲理想层的模空间。我们使用Fourier-Mukai技术来扩展Arcara和Bertram的思想，将穿墙表示为Mukai翻转，并证明了模空间是射影的。
We prove that, for a natural class of Bridgeland stability conditions on $\mathbb{P}^1\times\mathbb{P}^1$ and the blow-up of $\mathbb{P}^2$ at a point, the moduli spaces of Bridgeland semistable objects are projective. Our technique is to find suitable regions of stability conditions with hearts that are (after "rotation") equivalent to representations of a quiver. The helix and tilting theory is well-behaved on Del Pezzo surfaces and we conjecture that this program (begun in arXiv:1203.0316) runs successfully for all Del Pezzo surfaces, and the analogous Bridgeland moduli spaces are projectiv…
Higher rank DT/PT wall-crossing in Bridgeland stability
本文证明在 Picard 秩为 1 的光滑射影三维簇上，Gieseker 模空间与 PT 模空间仅被一道墙隔开，实现高阶秩 DT/PT 对应的墙穿越，并证明仅有限多道墙使相关对象不稳定。
Bridgeland's stabilities on abelian surfaces
本文研究交换曲面上Bridgeland稳定条件的壁结构。特别地，我们将研究Picard数为1的阿贝尔曲面上秩为1的复形的模空间的壁的结构。
Let $X$ be a smooth projective threefold of Picard number one for which the generalized Bogomlov-Gieseker inequality holds. We characterize the limit Bridgeland semistable objects at large volume in the vertical region of the geometric stability conditions associated to $X$ in complete generality and provide examples of asymptotically semistable objects. In the case of the projective space and $ch^β(E)=(-R,0,D,0)$, we prove that there are only a finite number of nested walls in the $(α,s)$-plane. Moreover, when $R=0$ the only semistable objects in the outermost chamber are the 1-dimensional Gi…
We show that the construction of Bayer, Bertram, Macri and Toda gives rise to a Bridgeland stability condition on a principally polarized abelian threefold with Picard rank one by establishing their conjectural generalized Bogomolov-Gieseker inequality for certain tilt stable objects. We do this by proving that a suitable Fourier-Mukai transform preserves the heart of a particular conjectural stability condition. We also show that the only reflexive sheaves with zero first and second Chern classes are the flat line bundles.
Determinants, even instantons and Bridgeland stability
本文提出计算例外集合对应quiver区域的系统方法，证明2步复形表示的μ-稳定层具有Bridgeland稳定性，并应用于证明偶秩2瞬子层、瞬子丛及反常瞬子的Bridgeland稳定性，同时刻画其模空间在唯一实际墙附近的结构。
Negative Effective Divisors and Bridgeland Stability of Line Bundles on Surfaces
针对连通光滑复射影曲面，证明Arcara–Miles猜想的有效除子版本及其严格类比，揭示线丛半稳定失效由负有效除子检测，相关数值条件与整数标度稳定性等价。
We show that the conjectural construction proposed by Bayer, Bertram, Macrí and Toda gives rise to Bridgeland stability conditions for a principally polarized abelian three-fold with Picard rank one by proving that tilt stable objects satisfy the strong Bogomolov-Gieseker type inequality. This is done by showing any Fourier-Mukai transform gives an equivalence of abelian categories which are double tilts of coherent sheaves.
Polynomial Bridgeland Stable Objects and Reflexive Sheaves
在一个光滑的射影三倍，我们表明，只有两个同构类型的稳定对象的模相对于拜耳的标准多项式Bridgeland稳定性-的Gieseker稳定层的模和PT稳定对象的模-在以下假设下：没有两个稳定向量是共线的，和程度和秩的对象是互素。我们还描述了PT-稳定和双PT-稳定对象的模空间的交集与自反层的模之间的密切关系。
Interpolation, Bridgeland stability and monomial schemes in the plane
给定一个零维格式Z，高阶插值问题要求对与Z的理想层上同调正交的向量丛的斜率进行分类。本文对射影平面上的所有零维单项式格式都解决了这个问题。作为一个推论，我们得到详细的信息，稳定的基础轨迹上的因子的希尔伯特计划的点在平面上。本文证明了Bridgeland稳定流形中的壁面与Arcara-Bertram-Coskun-Huizenga单项式格式的有效锥的Mori腔室分解中的壁面之间的对应关系。我们确定的Harder-Narasimhan过滤理想层的单式方案合适的Bridgeland稳定性条件，因此，获得一个新的决议更适合于上同调计算比其他标准的决议，如最小的自由分解。
Stability of Gieseker stable sheaves on K3 surfaces in the sense of Bridgeland and some applications
证明了投射K3曲面$X$上的Gieseker稳定层相对于Bridgeland在$X$的导出范畴上的稳定性条件是稳定的，只要该稳定性条件在依赖于层的稳定性条件空间的显式子集中.此外，我们将给出结果的两个应用。作为这些应用的一部分，我们证明了Picard数为1的K3曲面上Gieseker稳定无挠的层的细模空间是$μ$ -稳定局部自由层的模空间，如果层的秩不是平方数.
We prove that minimal instanton bundles on a Fano threefold $X$ of Picard rank one and index two are semistable objects in the Kuznetsov component $\mathsf{Ku}(X)$, with respect to the stability conditions constructed by Bayer, Lahoz, Macrì and Stellari. When the degree of $X$ is at least $3$, we show torsion free generalizations of minimal instantons are also semistable objects. As a result, we describe the moduli space of semistable objects with same numerical classes as minimal instantons in $\mathsf{Ku}(X)$. We also investigate the stability of acyclic extensions of non-minimal instantons.
Projectivity and Birational Geometry of Bridgeland moduli spaces
在Bridgeland意义下，我们在稳定复形的每个模空间上构造了一族nef因子类。这个除数类随Bridgeland稳定性条件自然变化。对于K3曲面上的一般稳定性条件，我们证明了该类是充分的，从而推广了Minamide，Yanagida和Yoshioka的结果。我们的结果还系统地解释了桥地稳定性的跨壁性与模空间的极小模型程序之间的关系。我们给出了我们的方法的三个应用经典的模空间的层上的K3曲面：1。在Gieseker稳定层的模空间中，我们得到了一个仅依赖于K3格的充分锥中的区域。2.我们确定的nef锥的Hilbert计划的n个点上的K3表面的Picard秩1时，n是大的相比，属。3.我们验证了“Hassett-Tschinkel/Huybrechts/Sawon”猜想的存在性的双有理拉格朗日纤维的希尔伯特计划在一个新的家庭的情况下。
Projectivity of Moduli Spaces in the Higher-Rank DT/PT Correspondence
研究光滑射影三维流形上DT/PT墙交叉处模空间的射影性，通过构造全局生成的线丛并分析其分离点的能力，建立高秩PT稳定对象的模空间的射影性。
Walls and asymptotics for Bridgeland stability conditions on 3-folds
研究三维簇上Bridgeland稳定性条件的墙结构与渐进行为，证明Gieseker半稳定性等价于渐近半稳定性，并展示如何构造大规模墙。
We study the reduced Donaldson-Thomas theory of abelian threefolds using Bridgeland stability conditions. The main result is the invariance of the reduced Donaldson-Thomas invariants under all derived autoequivalences, up to explicitly given wall-crossing terms. We also present a numerical criterion for the absence of walls in terms of a discriminant function. For principally polarized abelian threefolds of Picard rank one, the wall-crossing contributions are discussed in detail. The discussion yield evidence for a conjectural formula for curve counting invariants by Bryan, Pandharipande, Yin,…
$Z$-critical connections and Bridgeland stability conditions
本文将全纯向量丛上的几何偏微分方程与 Bridgeland 稳定性条件联系起来，定义 $Z$-临界联络，并证明大体积极限下其存在等价于渐近 $Z$-稳定，还给出高秩 dHYM 解的首批例子。
Rank Two Sheaves With Maximal Third Chern Class on the Fano Threefold of Index 2 and Degree 5
针对指数2、次数5的Fano三维簇$X_5$，通过倾斜稳定性与Bridgeland稳定性理论，完成具极大第三陈类的秩2半稳定层分类并明确其模空间性质。
In this short note, we describe a problem in algebraic geometry where the solution involves Catalan numbers. More specifically, we consider the derived category of coherent sheaves on an elliptic surface, and the action of its autoequivalence group on its Bridgeland stability manifold. In solving an equation involving this group action, the generating function of Catalan numbers arises, allowing us to use asymptotic estimates of Catalan numbers to arrive at a bound for the solution set.
Rank Two Sheaves With Low Discriminant on the Fano Threefold of Index 2 and Degree 5
研究在Fano三次折面X_5上描述判别式较低的秩2sheaves的稳定性条件及最大陈类特性。
Let $(S,L)$ be a polarized abelian surface of Picard rank one and let $ϕ$ be the function which takes each ample line bundle $L'$ to the least integer $k$ such that $L'$ is $k$-very ample but not $(k+1)$-very ample. We use Bridgeland's stability conditions and Fourier-Mukai techniques to give a closed formula for $ϕ(L^n)$ as a function of $n$ showing that it is linear in $n$ for $n>1$. As a byproduct, we calculate the walls in the Bridgeland stability space for certain Chern characters.
A hyperbolic metric and stability conditions on K3 surfaces with ρ=1
本文在射影K3曲面的稳定性条件的（正规化）空间上引入了一个双曲度量$X$，它具有Picard秩$ρ(X) =1$ .我们证明了在正规化空间中，所有的墙都是关于双曲度量的测地线。此外，我们证明了双曲度量是如何帮助我们主要讨论三个主题。本文首先研究了所谓的Bridgeland猜想。在第二个主题中，我们证明了一个著名的Orlov定理没有整体Torelli定理。在第三个主题中，我们利用双曲度量给出了一个在大体积极限下稳定复形的显式例子。虽然Bridgeland猜想可能是众所周知的代数几何学家，我们想开始从审查它。
We realize explicit symmetries of Bridgeland stability conditions on any abelian threefold given by Fourier-Mukai transforms. In particular, we extend the previous joint work with Maciocia to study the slope and tilt stabilities of sheaves and complexes under the Fourier-Mukai transforms, and then to show that certain Fourier-Mukai transforms give equivalences of the stability condition hearts of bounded t-structures which are double tilts of coherent sheaves. Consequently, we show that the conjectural construction proposed by Bayer, Macri and Toda gives rise to Bridgeland stability conditions…
Weak stability conditions on coherent systems of genus four curves
本文研究四曲面相干系统的稳定性条件，证明其退化为节点三次三折面Kuznetsov组件上的稳定性条件。
Stability conditions for polarised varieties
本文为极化簇引入了Z稳定性概念及其解析对应物Z临界Kähler度量，证明了在大体积情形下渐近Z稳定且解析K半稳定的极化簇允许Z临界Kähler度量，并揭示了其与局部壁穿越的联系。
A counterexample to the Jordan-Hölder property for polarizable semiorthogonal decompositions
本文证明极化半正交分解不满足Jordan-Hölder性质，在Fukaya范畴与光滑射影簇导出范畴中给出反例，并构造了一个秩为正但不允许稳定条件的dg范畴。
This article is based on a talk given at the Kinosaki Symposium on Algebraic Geometry in 2015, about a work in progress. We describe a polarization on a derived equivalent abelian variety by using Fourier-Mukai theory. We explicitly formulate a conjecture which says certain Fourier-Mukai transforms between derived categories give equivalences of some hearts of Bridgeland stability conditions. We establish it for abelian surfaces, which is already known due to D. Huybrechts, and for abelian 3-folds. This generalizes the author's previous joint work with A. Maciocia on principally polarized abel…
We investigate an extension of a lower bound on the Waring (cactus) rank of homogeneous forms due to Ranestad and Schreyer. We show that for particular classes of homogeneous forms, for which a generalization of this method applies, the lower bound extends to the level of border (cactus) rank. The approach is based on recent results on tensor asymptotic rank.
Asymptotic rank bounds: a numerical census
本文通过数值方法计算张量的改进渐近秩界，验证了Strassen猜想并揭示了现有数值方法的限制。
We use wall-crossing in the Bridgeland stability manifold to systematically study the birational geometry of the moduli space $M_σ(\mathbf{v})$ of $σ$-semistable objects of class $\mathbf{v}$ for a generic stability condition $σ$ on an arbitrary Enriques surface $X$. In particular, we show that for any other generic stability condition $τ$, the two moduli spaces $M_τ(\mathbf{v})$ and $M_σ(\mathbf{v})$ are birational. As a consequence, we show that for primitive $\mathbf{v}$ of odd rank $M_σ(\mathbf{v})$ is birational to a Hilbert scheme of points. Similarly, in even rank we show that $M_σ(\mat…
Shimura modular curves and asymptotic symmetric tensor rank of multiplication in any finite field
得到了任意$\F_q$在任意有限域的有限扩张上乘法的对称张量秩的新的渐近界.在这个目标中，我们使用对称Chudnovsky型广义算法应用于一族Shimura模曲线定义在$\F_{q^2}$达到Drinfeld-Vladut界和下降的定义域$\F_q$上。
New uniform and asymptotic upper bounds on the tensor rank of multiplication in extensions of finite fields
我们获得了新的统一上界的（不一定对称）张量秩的乘法在有限域$\F_q$的任何素数或素数幂$q\geq2$的扩展，而且这些统一的界限导致新的渐近界以及。此外，我们还给出了纯粹的渐近界，这是更好地使用家庭的Shimura曲线定义在$\F_q$，与最佳比率的$\F_{q^t}$合理的地方，他们的属，其中$q^t$是一个广场。
Asymptotic geometry of non-abelian Hodge theory and Riemann--Hilbert correspondence, rank three $\widetilde{E}_6$ case
本文在三孔球面上证明秩三的几何 P=W 猜想，通过渐近阿贝尔化与等变方法给出相关映射的 WKB 展开，并分析其 Stokes 现象，从而刻画特征簇在无穷远处的拓扑。
The deformed Vortex equations and equivariant stability conditions
研究紧致黎曼曲面\(X\)上\(X\times \mathbb{P}^1\)上\(SU(2)\)等变全纯向量丛的高阶dHYM方程，对一类涡旋型向量丛建立其方程解与\(Z\)-稳定性等价关系，且\(Z\)-稳定性与布里奇兰德稳定性相关。
Some metric properties of spaces of stability conditions
证明了在较弱的条件下，三角化范畴T上的数值Bridgeland稳定性条件Stab（T）空间是完备的.特别是Stab（T）的一个满分支上的度量，其中通过Grothendieck群K（T）的有限秩商的中心电荷因子是完备的。作为例子，我们计算亏格大于1的光滑复射影曲线的数值稳定性条件空间上的度量，并证明在这种情况下，由复数的自然作用所得到的商Stab（T）/C等距于具有一半双曲度量的上半半平面.我们还做了两个观察的方式，心脏的变化，因为我们通过空间的稳定条件。首先，在稳定性条件空间的相同分量中的稳定性条件的心通过倾斜的有限序列来关联。其次，如果稳定性条件的收敛序列中的每一个具有相同的心脏，则极限稳定性条件的心脏通过右倾斜由此获得。
Asymptotic directions in the moduli space of curves
本文研究亏格g曲线模空间切丛中由Torelli映射第二基本形式零化的渐近方向，给出g≥4时的例子，并依据切方向秩与曲线Clifford指标的关系给出渐近性判别条件，完整确定秩1并几乎完整描述秩2的渐近方向。
In connection with recent work on gaps in the asymptotic subranks of complex tensors the question arose whether the number of nonnegative real numbers that arise as the asymptotic subrank of some complex tensor is countable. In this short note we settle this question in the affirmative, for all tensor invariants that are algebraic in the sense that they are invariant under field automorphisms of the complex numbers.
We compute, via motivic wall-crossing, the generating function of virtual motives of the Quot scheme of points on $\mathbb{A}^3$, generalising to higher rank a result of Behrend, Bryan and Szendrői. We show that this motivic partition function converges to a Gaussian distribution, extending a result of Morrison.
Asymptotic Behavior of Tropical Rank Functions
本文研究了热带曲线上的两个主要秩概念的渐进行为，引入了与除子和热带模相关的热带体积概念，并证明了每种情况下的最优渐进行果，同时展示了热带体积与曲线的热带化相容性。
We make a first geometric study of three varieties in $\mathbb{C}^m \otimes \mathbb{C}^m \otimes \mathbb{C}^m$ (for each $m$), including the Zariski closure of the set of tight tensors, the tensors with continuous regular symmetry. Our motivation is to develop a geometric framework for Strassen's Asymptotic Rank Conjecture that the asymptotic rank of any tight tensor is minimal. In particular, we determine the dimension of the set of tight tensors. We prove that this dimension equals the dimension of the set of oblique tensors, a less restrictive class introduced by Strassen.
Systolic inequalities for K3 surfaces via stability conditions
本文引入Bridgeland稳定性条件的范畴收缩与范畴体积概念，证明任意射影K3曲面上存在仅依赖Picard群秩与判别式的常数C，使收缩平方不超过C倍体积，推广了二维环面经典收缩不等式并讨论其在辛几何中的应用。
Asymptotic tensor rank is characterized by polynomials
本文证明渐近张量秩是“从上可计算的”，即存在算法通过有限多项式列表判断其是否不超过给定值，并揭示其子水平集为Zariski闭集，从而推出渐近张量秩取值构成良序集。
Let $\operatorname{r_{max}}(n,d)$ be the maximum Waring rank for the set of all homogeneous polynomials of degree $d>0$ in $n$ indeterminates with coefficients in an algebraically closed field of characteristic zero. To our knowledge, when $n,d\ge 3$, the value of $\operatorname{r_{max}}(n,d)$ is known only for $(n,d)=(3,3),(3,4),(3,5),(4,3)$. We prove that $\operatorname{r_{max}}(3,d)=d^2/4+O(d)$ as a consequence of the upper bound $\operatorname{r_{max}}(3,d)\le\left\lfloor\left(d^2+6d+1\right)/4\right\rfloor$.
Let $X$ be a compact Riemann surface. Let $(E,θ)$ be a stable Higgs bundle of degree $0$ on $X$. Let $h_{\det(E)}$ denote a flat metric of the determinant bundle $\det(E)$. For any $t>0$, there exists a unique harmonic metric $h_t$ of $(E,θ)$ such that $\det(h_t)=h_{\det(E)}$. We prove that if the Higgs bundle is induced by a line bundle on the normalization of the spectral curve, then the sequence $h_t$ is convergent to the naturally defined decoupled harmonic metric at the speed of the exponential order. We also obtain a uniform convergence for such a family of Higgs bundles.
These lecture notes are intended as an introduction to several notions of tensor rank and their connections to the asymptotic complexity of matrix multiplication. The latter is studied with the exponent of matrix multiplication, which will be expressed in terms of tensor (border) rank, (border) symmetric rank and the asymptotic rank of certain tensors. We introduce the multilinear rank of a tensor as well, deal with the concept of tensor equivalence and study prehomogeneous vector spaces with the castling transform. Moreover, we treat Apolarity Theory and use it to determine the symmetric rank…
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