Lieux de dégénérescence (Degeneracy loci)
O. Debarre
证明了向量丛映射的退化轨迹的上同调群和Picard群的Lefschetz型定理。我们也处理反对称映射的情况。
We prove Lefschetz type theorems for cohomology groups and Picard groups of degeneracy loci for vector bundle maps. We also treat the case of antisymmetric maps.
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Théorèmes d'annulation et lieux de dégénérescence en petit corang
在给出了Grassmannian上的充量线丛的所有非零Dolbeault上同调群的明确刻画之后，给出了光滑射影簇上的充量向量丛的两个系列的消灭定理.它们暗示了富尔顿和拉扎斯菲尔德关于某些简并位点的连通性的猜想的一部分。
We give two formulas for the Chern-Schwartz-MacPherson class of symmetric and skew-symmetric degeneracy loci. We apply them in enumerative geometry, explore their algebraic combinatorics, and discuss K theory generalizations.
In [BFMT17] we introduced orbital degeneracy loci as generalizations of degeneracy loci of morphisms between vector bundles. Orbital degeneracy loci can be constructed from any stable subvariety of a representation of an algebraic group. In this paper we show that their canonical bundles can be conveniently controlled in the case where the affine coordinate ring of the subvariety is Gorenstein. We then study in a systematic way the subvarieties obtained as orbit closures in representations with finitely many orbits, and we determine the canonical bundles of the corresponding orbital degeneracy…
Geometry of lines and degeneracy loci of morphisms of vector bundles
科拉多塞格雷发挥了主导作用的基础线几何。我们综述了最近关于向量丛态射的退化轨迹的一些结果，他在这些结果中仍然具有深刻的启发意义。
We explain a general construction of double covers of quadratic degeneracy loci and Lagrangian intersection loci based on reflexive sheaves. We relate the double covers of quadratic degeneracy loci to the Stein factorizations of the relative Hilbert schemes of linear spaces of the corresponding quadric fibrations. We give a criterion for these double covers to be nonsingular. As applications of these results, we show that the double covers of the EPW sextics obtained by our construction give O'Grady's double EPW sextics and that an analogous construction gives Iliev-Kapustka-Kapustka-Ranestad'…
Let $S$ be the first degeneracy locus of a morphism of vector bundles corresponding to a general matrix of linear forms in $\mathbb{P}^s$. We prove that, under certain positivity conditions, its Hilbert square $\mathrm{Hilb}^2(S)$ is isomorphic to the zero locus of a global section of an irreducible homogeneous vector bundle on a product of Grassmannians. Our construction involves a naturally associated Fano variety, and an explicit description of the isomorphism.
We prove that the Hilbert scheme of degeneracy loci of pairs of global sections of Omega(2), the twisted cotangent bundle on P^(n-1), is unirational and dominated by the Grassmannian of lines in the projective space of skew-symmetric forms over a vector space of dimension n. We provide a constructive method to find the fibers of the dominant map. In classical terminology, this amounts to giving a method to realize all the pencils of linear line complexes having a prescribed set of centers. In particular, we show that the previous map is birational when n=4.
On the Hilbert scheme of degeneracy loci of twisted differential forms
证明了当3 < m < n-1时，n维向量空间上的斜对称形式空间的m维子空间的格拉斯曼算子对于P^{n-1}上的扭余切丛Ω（2）的m个整体瓣的退化轨迹的Hilbert概型是双有理的.当3=m<n-1且n为奇数时，证明了该Grassmannian对于由n阶线性反对称矩阵的Pfronese曲面参数化的Veronese曲面集是双有理的.
Degeneracy loci of morphisms between vector bundles have been used in a wide variety of situations. We introduce a vast generalization of this notion, based on orbit closures of algebraic groups in their linear representations. A preferred class of our orbital degeneracy loci is characterized by a certain crepancy condition on the orbit closure, that allows to get some control on the canonical sheaf. This condition is fulfilled for Richardson nilpotent orbits, and also for partially decomposable skew-symmetric three-forms in six variables. In order to illustrate the efficiency and flexibility …
Using raising operators and geometric arguments, we establish formulas for the K-theory classes of degeneracy loci in classical types. We also find new determinantal and Pfaffian expressions for classical cases considered by Giambelli: the loci where a generic matrix drops rank, and where a generic symmetric or skew-symmetric matrix drops rank. In an appendix, we construct a K-theoretic Euler class for even-rank vector bundles with quadratic form, refining the Chow-theoretic class introduced by Edidin and Graham. We also establish a relation between top Chern classes of maximal isotropic subbu…
Versality of Brill-Noether flags and degeneracy loci of twice-marked curves
本文研究一般双标记曲线上Brill-Noether退化轨迹的几何性质，通过证明一对旗的通用性定理，确定了其维数、奇异轨迹和相交类，并给出相交类的组合解释。
Given a vector bundle on a $\mathbb{P}^1$ bundle, the base is stratified by degeneracy loci measuring the spitting type of the vector bundle restricted to each fiber. The classes of these degeneracy loci in the Chow ring or cohomology ring of the base are natural invariants characterizing the degenerations of the vector bundle. When these degeneracy loci occur in the expected codimension, we find their classes. This yields universal formulas for degeneracy classes in terms of naturally arising vector bundles on the base. Our results hold over arbitrary fields of any characteristic.
The Degeneracy Loci for Smooth Moduli of Sheaves
本文研究了稳定sheaves模空间上universal sheaf的退化领域，证明其为不可约Cohen-Macaulay品种，并推广了Hilbert方案的相关工作。
Nonemptiness of symmetric degeneracy loci
设V是d维复射影概形X上的秩N向量丛;设V具有值在线丛L中的二次型，且S ^2 V^* \otimes L是充足的。设二次型在X中任意点的最大秩为r > 0。本文的主要结果是：当d > N-r时，二次型的秩至多为r-1的点的轨迹非空。本文给出了矩阵的子概型和与射影空间中嵌入有关的退化轨迹的一些应用。本文最后附录Gysin地图。附录中的主要结果证明了一个Gysin映射与自然映射从普通上同调到相对上同调。
On a conjecture on aCM and Ulrich sheaves on degeneracy loci
本文针对Kleppe与Miró-Roig关于标准行列式轨迹法层外幂扭曲后为算术Cohen–Macaulay（线性行列式情形为Ulrich）的猜想，利用Weyman几何方法给出这类层的简单局部自由分解以证明猜想。
We study equivariant bi-vector fields on a toric variety. We prove that, on a smooth toric variety of dimension $n$, the locus where the rank of an equivariant bi-vector field is $\leq 2k$ is not empty and has at least a component of dimension $\geq 2k+1$, for all integers $k> 0$ such that $2k < n$. The same is true also for $k=0$, if the toric variety is smooth and compact. While for the non compact case, the locus in question has to be assumed to be non empty.
The Coble cubics were discovered more than a century ago in connection with genus two Riemann surfaces and theta functions. They have attracted renewed interest ever since. Recently, they were reinterpreted in terms of alternating trivectors in nine variables. Exploring this relation further, we show how the Hilbert scheme of pairs of points on an abelian surface, and also its Kummer fourfold, a very remarkable hyper-Kähler manifold, can very naturally be constructed in this context. Moreover, we explain how this perspective allows us to describe the group law of an abelian surface, in a strik…
On the connectedness of some degeneracy loci and of Ulrich subvarieties
本文研究了某些退化 locus 和 Ulrich 子变种的连通性，通过 Chern 类的消失条件和 V-大性证明了其连通性，并在 Ulrich 包情况下给出了更详细的结果。
Degeneracy Loci, Pfaffians, and Vexillary Signed Permutations in Types B, C, and D
定义了B、C、D型向量丛的符号置换的概念，对应于具有这些对称性的向量丛的自然退化轨迹。我们表明，这些轨迹的类给出明确的Pfweian公式。Kazarian的Grassmannian公式是重要的特例，而池田、Mihalcea和Naruse的相应的双Schubert多项式被证明等于这些Pfuberans。
Nonemptiness of skew-symmetric degeneracy loci
设V是d维复射影概形X上的秩N向量丛;设V具有反称双线性型，其值在线丛L中，且Λ^2 V^* \otimes L是充足的。假设形式在X的任意点的最大秩是r，其中r>0是偶数。本文的主要结果是：若d>2（N-r），则其秩至多为r-2的点的轨迹非空。这是数学AG/0305159的主要结果的斜对称模拟;证明是类似的。如果放宽了对满度的假设，我们得到了X的最大维数的一个较弱的估计（并给出了对称情况下的类似结果）。我们给应用程序的反对称矩阵的子计划，并分层的对偶的李代数的轨道维数。
Motivic Classes of isotropic degeneracy loci and symmetric orbit closures
本文提供计算交错退化区域的动机柴恩和希泽布鲁赫类的显式公式，特别涉及辛和奇正交格拉斯曼流形。结果扩展了Anderson-Chen-Tarasca在普通格拉斯曼退化区域的研究。
Double Schubert polynomials and degeneracy loci for the classical groups
本文提出了B，C，D型李的双Schubert多项式P_w（X，Y）理论，它自然地推广了A型Schutzenberger的Lascoux多项式族。这些多项式满足正性、正交性和稳定性，并代表了向量丛的Schubert簇和退化轨迹类。当w是外尔群的极大Grassmannian元时，P_w（X，Y）可以用Schur型行列式和Pfirmans表示，类似于Kempf和Laksov的A型公式.由量子上同调激发的一个例子表明，丛的“各向同性态射”的退化轨迹不存在陈类公式。
We consider a generalized Riemann-Hurwitz formula as it may be applied to rational maps between projective varieties having an indeterminacy set and fold-like singularities. The case of a holomorphic branched covering map is recalled. Then we see how the formula can be applied to iterated maps having branch-like singularities. Separately, we consider a further application involving the Chern classes of determinantal varieties when the latter are realized as the degeneracy loci of certain vector bundle morphisms.
Recent developments on the uniformity of the number of rational points on curves and subvarieties in a moving abelian variety rely on the geometric concept of the degeneracy locus. The first-named author investigated the degeneracy locus in certain mixed Shimura varieties. In this expository note we revisit some of these results while minimizing the use of mixed Shimura varieties while working in a family of principally polarized abelian varieties. We also explain their relevance for applications in diophantine geometry.
Motivic Chern and Hirzebruch classes are polynomials with K-theory and homology classes as coefficients, which specialize to Chern-Schwartz-MacPherson classes, K-theory classes, and Cappell-Shaneson L-classes. We provide formulas to compute the motivic Chern and Hirzebruch classes of Grassmannian and vexillary degeneracy loci. We apply our results to obtain the Hirzebruch $χ_y$-genus of classical and one-pointed Brill-Noether varieties, and therefore their topological Euler characteristic, holomorphic Euler characteristic, and signature.
The Chern-Schwartz-MacPherson class (CSM) and the Segre-Schwartz-MacPherson class (SSM) are deformations of the fundamental class of an algebraic variety. They encode finer enumerative invariants of the variety than its fundamental class. In this paper we offer three contributions to the theory of equivariant CSM/SSM classes. First, we prove an interpolation characterization for CSM classes of certain representations. This method---inspired by recent works of Maulik-Okounkov and Gorbounov-Rimanyi-Tarasov-Varchenko---does not require a resolution of singularities and often produces explicit (no…
In this paper, we prove determinant formulas for the $K$-theory classes of the structure sheaves of degeneracy loci classes associated to vexillary permutations in type $A$. As a consequence we obtain determinant formulas for Lascoux-Schützenberger's double Grothendieck polynomials associated to vexillary permutations. Furthermore, we generalize the determinant formula to algebraic cobordism.
We give an explicit expression for the contact loci of hyperplane arrangements and show that their cohomology rings are combinatorial invariants. We also give an expression for the restricted contact loci in terms of Milnor fibers of associated hyperplane arrangements. We prove the degeneracy of a spectral sequence related to the restricted contact loci of a hyperplane arrangement and which conjecturally computes algebraically the Floer cohomology of iterates of the Milnor monodromy. We give formulas for the Betti numbers of contact loci and restricted contact loci in generic cases.
This paper originated as an attempt to answer a question: what are the natural derived structures on Poisson degeneracy loci? We argue that the question could be possibly answered via a construction of differential graded operads that ``naturally'' act on the degeneracy loci. For each $m \ge 0$, we suggest what looks like a reasonable condition for a Poisson structure on a commutative differential graded algebra to be $m$-degenerate, i.e. to ``have rank $\le 2m$''. That condition will turn out to be a universal property of the operad that controls such Poisson algebras; we denote that operad $…
Degeneracy loci and polynomial equation solving
设V是复数$C$上维数为r的光滑等维拟仿射簇，$F$是$C[V]$的坐标函数的$(p\times s)$ -矩阵，其中$s\ge p+r$ .对$(V,F)$确定秩为$s-p$除以$W:=\{x\in V:\mathrm{rk} F(x)=p\}$的向量束$E$。我们与$(V,F)$一个下降链的简并位点的E（通用极性品种的$V$代表一个典型的例子，这种情况）。这些退化轨迹的最大程度构成了我们将要设计的统一的、有界误差概率伪多项式时间算法的基本成分，并且解决了可以在这个框架中制定的一系列计算消除问题。我们描述的应用程序，多项式方程解决实数和计算的一个通用纤维的一个占主导地位的自同态的仿射空间。
We express nested Hilbert schemes of points and curves on a smooth projective surface as "virtual resolutions" of degeneracy loci of maps of vector bundles on smooth ambient spaces. We show how to modify the resulting obstruction theories to produce the virtual cycles of Vafa-Witten theory and other sheaf-counting problems. The result is an effective way of calculating invariants (VW, SW, local PT and local DT) via Thom-Porteous-like Chern class formulae.
Formulas for Lagrangian and orthogonal degeneracy loci; the Q-polynomials approach
设V是赋非退化辛或正交形式的概型X上的向量丛。设G是V的极大迷向子丛的参数化的Grassmannian丛，本文的主要目的是给出G中的轨迹类的公式，这些轨迹类的参数化子丛与V的迷向子丛的固定旗的连续成员相交，维数大于1，2，3，…. .我们给出的公式是G上重言丛的Q-多项式和旗成员的旗S-多项式的二次表达式。这些公式全球化的拉格朗日和正交退化轨迹。它们对于X中两个极大迷向子丛相交维数超过给定数的点的轨迹有特别好的形式。（几年前，哈里斯提出了一个寻找这些轨迹公式的问题。）在这种情况下，我们的公式是应用于E和F的Q多项式的简单二次表达式。本文推广了[P.Pragacz，Schur S-和Q-多项式的代数几何应用，SLN 1478（1991），130-191]中给出的公式。其应用之一是计算Pryms中Brill-Noether基因座的类别，见[De Concini，P. Pragacz，On the class of Brill-Noether loci for Prym varieties，Math.Ann.302（1995），687-697]。
In previous work, we employed a geometric method of Kazarian to prove Pfaffian formulas for a certain class of degeneracy loci in types B, C, and D. Here we refine that approach to obtain formulas for more general loci, including those coming from all isotropic Grassmannians. In these cases, the formulas recover the remarkable theta- and eta-polynomials of Buch, Kresch, Tamvakis, and Wilson. The streamlined geometric approch yields simple and direct proofs, which proceed in parallel for all four classical types. In an appendix, we develop some foundational algebra and prove several Pfaffian id…
Poisson modules and degeneracy loci
本文研究了全纯Poisson几何中模与子对象之间的相互作用。特别是，我们定义了一个新的概念“残留”的泊松模，类似于庞加莱残留的亚纯体积形式。特别感兴趣的是泊松流形的规范线丛和它的退化轨迹的残留物之间的相互作用-在那里的泊松结构的秩下降。作为一个应用程序，我们提供了新的证据支持Bondal的猜想，秩\leq 2k轨迹的Fano泊松流形总是有尺寸\geq 2k+1。特别是，我们表明，猜想持有Fano fourfolds。我们也将我们的技术应用到一个家庭的泊松结构定义的Fe \u ıgin和Odesski \u的，其中的退化轨迹给出的椭圆正态曲线的正割品种。
Schubert complexes and degeneracy loci
给定Cohen-Macaulay簇上标记向量丛之间的通有映射，构造了Schubert型退化轨迹上具有线性分解的极大Cohen-Macaulay模.线性分辨率由Schubert复形提供，这是本文介绍和研究的主要工具。这些复形扩展了Krakenkiewicz和Pragacz的Schubert函子，并受到Schur复形分解行列式簇上支持的极大Cohen-Macaulay模的事实的启发。在K理论中得到的公式提供了简并轨迹的结构层的“线性逼近“，它可以用来恢复富尔顿的公式。
Given a map of vector bundles on a smooth variety, consider the deepest degeneracy locus where its rank is smallest. We show it carries a natural perfect obstruction theory whose virtual cycle can be calculated by the Thom-Porteous formula. We show nested Hilbert schemes of points on surfaces can be expressed as degeneracy loci. We show how to modify the resulting obstruction theories to recover the virtual cycles of Vafa-Witten and reduced local DT theories. The result computes some Vafa-Witten invariants in terms of Carlsson-Okounkov operators. This proves and extends a conjecture of Gholamp…
K-orbit closures on G/B as universal degeneracy loci for flagged vector bundles with symmetric or skew-symmetric bilinear form
我们使用等变的本地化和均差算子，以确定公式的环面等变的基本上同调类的$K$ -轨道闭包的标志品种$G/B$，其中$G = GL(n,\C)$，其中$K$是一个对称子群$O(n,\C)$或$Sp(n,\C)$。我们实现这些轨道闭包作为普遍退化轨迹的向量丛在各种配备了一个单一的标志的子丛和非退化对称或反称双线性型的价值观在平凡的捆绑。我们描述了如何我们的等变公式可以被解释为给出公式的类这样的轨迹的陈类的各种束。
Hecke cycles on moduli of vector bundles and orbital degeneracy loci
本研究从一般三向量出发构造轨道退化轨迹型法诺流形，证明其定义亏格二曲线的秩3半稳定向量丛模空间中的Hecke线族，建立其与奇模空间的同构关系，并得到亏格19的K3曲面。
Real degeneracy loci of matrices and phase retrieval
本文受相位恢复启发，研究实空间上线性算子的退化轨迹，刻画了三维空间中四算子退化轨迹的四边形构型条件，并推广得到对应Sylvester谱面体的广义Desargues构型结果。
We prove a determinantal formula and Pfaffian formulas that respectively describe the $K$-theoretic degeneracy loci classes for Grassmann bundles and for symplectic Grassmann and odd orthogonal bundles. The former generalizes Damon--Kempf--Laksov's determinantal formula and the latter generalize Pragacz--Kazarian's formula for the Chow ring. As an application, we introduce the factorial $GΘ/ GΘ'$-functions representing the torus equivariant $K$-theoretic Schubert classes of the symplectic and the odd orthogonal Grassmannians, which generalize the (double) theta polynomials of Buch--Kresch--Tam…
Symmetric subgroup orbit closures on flag varieties: Their equivariant geometry, combinatorics, and connections with degeneracy loci
当$G$是典型群$SL(n,\C)$、$SO(n,\C)$或$Sp(2n,\C)$之一，且$K$是$G$的对称子群时，给出了标志簇$G/B$上闭$K$ -轨道的环面等变基本类的显式公式.我们描述的参数化的每个轨道集和组合的弱顺序，使我们能够计算公式的等变类的所有剩余的轨道封闭使用均差运营商。在每一种情况下，在类型A，我们实现了$K$ -轨道闭包的某种类型的通用退化轨迹，涉及一个向量丛$V$的计划$X$配备了一个标志的子丛和进一步的结构由$K$。我们描述了如何我们的等变公式可以被解释为这样的轨迹在陈类的各种束$X$的公式。
Differential complexes and Hodge theory on log-symplectic manifolds
本文研究对数辛流形上的微分形式复形（含反向de Rham复形），将其与Poisson双矢量的退化除子及秩轨迹关联，并在Kähler情形下建立重数轨迹与Hodge数的关系，同时揭示Hodge数消失与无阻碍形变的联系。
Two-pointed Prym-Brill-Noether Loci and coupled Prym-Petri theorem
本文建立了具有特殊消没的双点Prym-Brill-Noether轨迹，利用D型vexillary退化轨迹的K-理论公式确定其K-理论类，并借助耦合Prym-Petri映射证明了双点Prym-Petri定理，推广了前人的无点和带点情形。
K-orbit closures on G/B as universal degeneracy loci for flagged vector bundles splitting as direct sums
我们使用等变本地化和除差算子，以确定公式的环面等变基本上同调类的$K$ -轨道封闭的旗品种$G/B$的各种对称对$(G,K)$。在$A$型中，我们将$K=GL(p,\C) \times GL(q,\C)$ -轨道在$GL(p+q,\C)/B$上的闭包实现为一个簇上向量丛的泛退化轨迹，该簇具有一个单一的子丛标记，并且分裂为秩$p$和$q$的子丛的直和.这种退化轨迹的精确描述依赖于知道一个集合理论描述的$K$ -轨道封闭，我们提供了通过一个详细的组合分析的偏序集“$(p,q)$ -氏族”，参数化的轨道封闭。我们精确地描述了我们的公式的等变类的$K$轨道封闭可以被解释为公式的类，这样的退化轨迹的陈类所涉及的束。在类型$A$之外的情况下，我们建议轨道闭包应该参数化退化轨迹，涉及配备有非退化对称或反对称双线性形式的向量丛，相对于该形式是各向同性或拉格朗日的子丛的单个标志，以及作为每个被加数满足相对于该形式的某些性质（取决于$K$）的子丛的直和的分裂。精确描述这样一个简并位点的所有情况下，在类型$B$和$C$。
In our previous work arXiv:1305.3543, we employed the approach to Schubert polynomials by Fomin, Stanley, and Kirillov to obtain simple, uniform proofs that the double Schubert polynomials of Lascoux and Schutzenberger and Ikeda, Mihalcea, and Naruse represent degeneracy loci for the classical groups in the sense of Fulton. Using this as our starting point, and purely combinatorial methods, we obtain a new proof of the general formulas of arXiv:0908.3628, which represent the degeneracy loci coming from any isotropic partial flag variety. Along the way, we also find several new formulas and elu…
Quiver polynomials in iterated residue form
表示的退化轨迹多项式推广了代数组合学中几个重要的多项式。本文给出了这类多项式的非常规生成序列描述，当多项式为Dynkin型时。
In this paper we provide generalisations of Pfaffian formulas for the degeneracy loci classes in the algebraic cobordism of symplectic/odd orthogonal Grassmann bundles.
On symmetric degeneracy loci, spaces of symmetric matrices of constant rank and dual varieties
设X是m维非奇异单连通射影簇，E是X上的秩n向量丛，L是X上的线丛。设$S^2(E^{*}) \otimes L$是一个充分向量丛，且存在一个常数偶数秩$r \ge 2$对称丛映射$E \to E^{*} \otimes L$ .证明了$m \le n-r$ .利用这一结果解决了对称矩阵的常秩问题，证明了$m\times m$对称矩阵空间的线性子空间的最大维数是$m-r+1$，使得每个非零元素都有偶数秩$r \ge 2$.我们解释了这个结果如何涉及到研究的对偶簇在射影几何，并给出了一些应用和例子。
Degeneracy loci and unlikely intersections in abelian schemes
本文证明在阿贝尔簇的正维子簇中，除退化轨迹外，相对维数小于给定阈值的平坦群子簇中的点至多有限，并由此提出并部分证明了相对 Mordell--Lang 猜想。
Chern class formulas for $G_2$ Schubert loci
本文定义了具有结构群$G_2$的向量丛的退化轨迹，并给出了它们的上同调（或Chow）类的公式.当基是点时，这些公式是Bernstein-Gelfand-Gelfand和Demazure发展的有理齐性空间理论的一部分。这已被扩展到设置一般代数几何的Giambelli-Thom-Porteous，肯普夫-Laksov，和富尔顿在经典类型;目前的工作进行类似的计划类型$G_2$。我们包括明确的描述的$G_2$标志品种和舒伯特品种，和几个计算，包括一个回答问题的W。格雷姆在附录中，我们从表示论中收集了一些事实，并计算了二次丛的Chow环，澄清了Edidin和Graham以前的计算。
The Poisson degeneracy locus of a flag variety
本研究针对配备标准泊松结构的复半单李群完全旗簇的泊松退化轨迹展开系统分析，给出组合描述与结构刻画，证明其不可约分支同构于复射影直线的幂次，并将结果推广至部分旗簇，明确了最高维数的组合与代数表示论意义。
By analyzing degeneracy loci over projectivized vector bundles, we recompute the degree of the discriminant locus of a vector bundle and provide a new proof of the Bogomolov instability theorem.
Probability-theoretic interpretation of degeneracy locus formulas
本文用随机有理六顶点模型解释退化轨迹的Schwartz-MacPherson类，完成三种经典类型的稳定SSM理论，将斜类与对称类分别解释为概率和Chebyshev矩。
Universal Schubert polynomials
我们引入了表示向量丛映射的一般退化轨迹的多项式。这些多项式专门用于已知的经典和量子形式的单和双舒伯特多项式。这是该论文的最终版本，发表在《杜克大学数学杂志》上。结果得到了加强，因为拉克什米拜和马扎尔证明了一个来自前一版本的猜想：在普遍情况下，这些轨迹是科恩-麦考利。
Degeneracy locus formulas for amenable Weyl group elements
本文提出了一类可容Weyl群元素，并推导了其在辛和正交退化领域中的上同调类公式，推广了双theta和eta多项式。
We prove a Pfaffain formula for the $K$-theory class of the degeneracy loci in the bundle of odd maximal orthogonal Grassmannians.
Moduli of Abelian varieties, Vinberg theta-groups, and free resolutions
我们提出了一个系统的方法来研究几何方面的温伯格θ-表示。其主要思想是利用Borel-Weil构造将约化群表示为齐性空间上齐性丛的截面，然后研究这些向量丛的退化轨迹.我们的主要技术工具是使用自由分解作为简并轨迹公式的“增强”版本。我们说明我们的方法上的几个例子，并显示它们是如何连接到模空间的阿贝尔簇。为了使这篇文章访问代数学家和geometers，我们还包括背景材料的自由分解和表示论。
Vanishing theorems for ample vector bundles
证明了光滑复射影簇上的充量向量丛的对称幂和反对称幂乘积的上同调的一般消失定理。特殊情况包括一个扩展的经典定理格里菲斯和乐Potier整个Dolbeault上同调，并回答一个问题提出的Demailly。应用于退化轨迹。
On the construction of some Buchsbaum varieties and the Hilbert scheme of elliptic scrolls in P^5
本文从$P^n$到$Ω(2)$的结构层的m个拷贝的直和，以及从将$Ω(2)$的截面作为线性线聚的经典几何解释的观点出发，研究了一般丛态射的退化轨迹.我们特别考虑m= 2，3的$P^5$的情况。当n=5，m=3时，通过定义一个从Grassmannian G（2，14）到H的自然有理映射，给出了$P^5$中椭圆法向涡卷的Hilbert概型H的一个显式描述，其结果是具有一般四次纤维的优势.
Let G be a classical complex Lie group, P any parabolic subgroup of G, and X = G/P the corresponding homogeneous space, which parametrizes (isotropic) partial flags of subspaces of a fixed vector space. In the mid 1990s, Fulton, Pragacz, and Ratajski asked for global formulas which express the cohomology classes of the universal Schubert varieties in flag bundles -- when the space X varies in an algebraic family -- in terms of the Chern classes of the vector bundles involved in their definition. This has applications to the theory of degeneracy loci of vector bundles and is closely related to …
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