Robert Laterveer
Let $Y$ be a Plücker hypersurface in a symplectic Grassmannian $I_1 Gr(3,n)$ or a bisymplectic Grassmannian $I_2 Gr(3,n)$. We show that many Chow groups of $Y$ inject into cohomology.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
继续探索
Motivated by the generalized Bloch conjecture, we formulate a conjecture about the Chow groups of Plücker hypersurfaces in Grassmannians. We prove weak versions of this conjecture.
This small note is about Plücker hyperplane sections $X$ of the Grassmannian $\operatorname{Gr}(3,V_{10})$. Inspired by the analogy with cubic fourfolds, we prove that the only non-trivial Chow group of $X$ is generated by Grassmannians of type $\operatorname{Gr}(3,W_{6})$ contained in $X$. We also prove that a certain subring of the Chow ring of $X$ (containing all intersections of positive-codimensional subvarieties) injects into cohomology.
Chow-Witt rings of Grassmannians
本文计算了一般线性群分类空间的Chow-Witt环，涉及非平凡对偶性，并以此推导出有限Grassmannian的Chow-Witt环，所得公式与实Grassmannian整上同调环公式高度相似。
We extend a result of to Esnault-Levine-Viehweg concerning the Chow groups of hypersurfaces in projective space to those in weighted projective spaces.
In this appendix we tabulate preliminary data used in establishing Theorem 1-14 in the paper ``The Chow rings of generalized Grassmannians''
The literature on maximal torus orbits in the Grassmannian is vast; in this paper we initiate a program to extend this to diagonal subtori. Our main focus is generalizing portions of Kapranov's seminal work on Chow quotient compactifications of these orbit spaces. This leads naturally to discrete polymatroids, generalizing the matroidal framework underlying Kapranov's results. By generalizing the Gelfand-MacPherson isomorphism, these Chow quotients are seen to compactify spaces of arrangements of parameterized linear subspaces, and a generalized Gale duality holds here. A special case is birat…
We study the product structure on the Chow ring (with rational coefficients) of a cubic hypersurface in projective space and prove that the image of the product map is as small as possible.
For any algebraic scheme $X$ and every $(n,\mathscr{L})\in \mathbb{Z}\times \text{Pic}(X)$ we define an associated involution of its Chow group $A_*X$, and show that certain characteristic classes of (possibly singular) hypersurfaces in a smooth variety are interchanged via these involutions. For $X=\mathbb{P}^N$ we show that such involutions are induced by involutive correspondences.
We apply the previous calculations of Chow-Witt rings of Grassmannians to develop an oriented analogue of the classical Schubert calculus. As a result, we get complete diagrammatic descriptions of the ring structure in Chow-Witt rings and twisted Witt groups. In the resulting arithmetic refinements of Schubert calculus, the multiplicity of a solution subspace is a quadratic form encoding additional orientation information. We also discuss a couple of applications, such as a Chow-Witt version of the signed count of balanced subspaces of Fehér and Matszangosz.
We prove the existence of a canonical zero-cycle on a Calabi-Yau hypersurface X in a complex projective homogeneous variety. More precisely, we show that the intersection of any n divisors on X, n=dim X, is proportional to the class of a point on a rational curve in X.
Chow-Lam Recovery
本文研究 Grassmannian 子簇能否由其 Chow-Lam 形式及线性投影恢复，给出恢复的必要条件，并构造与射影空间情形相反、无法恢复的反例族。
We consider Kapranov's Chow quotient compactification of the moduli space of ordered n-tuples of hyperplanes in P^{r-1} in linear general position. For r=2 this is canonically identified with the Grothendieck-Knudsen compactification of M_{0,n} which has among others the nice properties 1) Modular meaning: stable pointed rational curves 2) Canonical description of limits of one parameter degenerations 3) Natural Mori theoretic meaning: log canonical compactification. We prove (1-2) generalize naturally to all (r,n), but that (3), which we view as the deepest, fails except possibly in the cases…
On one hand, for a general Calabi-Yau complete intersection X, we establish a decomposition, up to rational equivalence, of the small diagonal in X^3, from which we deduce that any decomposable 0-cycle of degree 0 is in fact rationally equivalent to 0, up to torsion. On the other hand, we find a similar decomposition of the smallest diagonal in a higher power of a hypersurface, which provides us an analogous result on the multiplicative structure of its Chow ring.
Let M_{P^k}(P^r, d) be the moduli space of unparameterized maps μ:P^k -> P^r satisfying μ^*(O(1))= O(d). M_{P^k}(P^r,d) is a quasi-projective variety, and, in case k=1, M_{P^1}(P^r,d) is the fundamental open cell of Kontsevich's space of stable maps \bar{M}_{0,0}(P^r,d). It is shown that the Q-coefficient Chow ring of M_{P^k}(P^r,d) is canonically isomorphic to the Chow ring of the Grassmannian Gr(P^k, P^r)= M_{P^k}(P^r,1).
To every projective variety $X$, we associate a family of hypersurfaces in different Grassmannians, called the coisotropic hypersurfaces of $X$. These include the Chow form and the Hurwitz form of $X$. Gel'fand, Kapranov and Zelevinsky characterized coisotropic hypersurfaces by a rank one condition on tangent spaces. We present a new and simplified proof of that result. We show that the coisotropic hypersurfaces of $X$ equal those of its projectively dual variety, and that their degrees are the polar degrees of $X$. Coisotropic hypersurfaces of Segre varieties are defined by hyperdeterminants,…
We introduce a certain compactification of the space of projective configurations i.e. orbits of the group $PGL(k)$ on the space of $n$ - tuples of points in $P^{k-1}$ in general position. This compactification differs considerably from Mumford's geometric invariant theory quotient. It is obtained by considering limit position (in the Chow variety) of the closures of generic orbits. The same result will be obtained if we study orbits of the maximal torus on the Grassmannian $G(k,n)$. We study in detail the closures of the torus orbits and their "visible contours" which are Veronese varieties i…
A short way of counting maps to hypersurfaces in Grassmannians
该研究利用Quot概形紧化计算从亏格g光滑射影曲线到格拉斯曼流形超曲面的d次映射虚计数，并探究大次数d下该虚计数在合适超曲面条件下是否具有枚举性。
Based on the Basis theorem of Bruhat--Chevalley [C] and the formula for multiplying Schubert classes obtained in [D\QTR{group}{u}] and programed in [DZ$_{\QTR{group}{1}}$], we introduce a new method computing the Chow rings of flag varieties (resp. the integral cohomology of homogeneous spaces). The method and results of this paper have been extended in [DZ$_{3}$, DZ$_{4}$] to obtain the integral cohomology rings of all complete flag manifolds, and to construct the integral cohomologies of Lie groups in terms of Schubert classes.
In this paper we further develop a Grassmannian technique used to prove results about very general hypersurfaces and present three applications. First, we provide a short proof of the Kobayashi Conjecture given previous results on the Green-Griffiths-Lang Conjecture. Second, we characterize the dimension of the space of Chow-equivalent points on a very general hypersurface, proving the remaining cases of a conjecture of Chen, Lewis and Sheng and providing a short, alternate proof for many of the already known cases. Finally, we relate Seshadri constants of very general points to Seshadri const…
In this paper, we prove formulas that represent two-pointed Gromov-Witten invariant <O_{h^a}O_{h^b}>_{0,d} of projective hypersurfaces with d=1,2 in terms of Chow ring of Mbar_{0,2}(P^{N-1},d), the moduli spaces of stable maps from genus 0 stable curves to projective space P^{N-1}. Our formulas are based on representation of the intersection number w(O_{h^a}O_{h^b})_{0,d}, which was introduced by Jinzenji, in terms of Chow ring of Mp_{0,2}(N,d), the moduli space of quasi maps from P^1 to P^{N-1} with two marked points. In order to prove our formulas, we use the results on Chow ring of Mbar_{0,…
We show that when $d \geq 3$, the Hilbert scheme $Hilb_{dT+1-\binom{d-1}{2}}(G(k,n))$ has 2 components, even though elements in both components have the same cohomology class. Moreover, we show that the Hilbert scheme associated to the Hilbert polynomial $\binom{T+n}{n}-\binom{T+n-d}{n}$ in Grassmannian has at most 2 connected components.
In this paper, we give a Prym-Tjurin construction for the cohomology and the Chow groups for a cubic hypersurface. On the space of lines meeting a given rational curve, there is the incidence correspondence. This correspondence induces an action on both of the primitive cohomology and the primitive Chow groups. We first show that this action satisfies a quadratic equation. Then the Abel-Jacobi homomorphism induces an isomorphism between the primitive cohomology of the cubic hypersurface and the Prym-Tjurin part of the above action. This also holds for Chow groups with rational coefficients.
We show that when $d \geq 3$ and $m>2$, the Nef cone of the Hilbert scheme $Hilb_{P_{d,m}(T)}(G(k,n))$ is a cone spanned by 6 classes in general case, where $P_{d,m}(T)=\binom{T+m}{m}-\binom{T+m-d}{m}$.
The most fundamental example of mirror symmetry compares the Fermat hypersurfaces in P^n and P^n/G, where G is a finite group that acts on P^n and preserves the Fermat hypersurface. We generalise this to hypersurfaces in Grassmannians, where the picture is richer and more complex. There is a finite group G that acts on the Grassmannian Gr(n,r) and preserves an appropriate Calabi-Yau hypersurface. We establish how mirror symmetry, toric degenerations, blow-ups and variation of GIT relate the Calabi-Yau hypersurfaces inside Gr(n,r) and Gr(n,r)/G. This allows us to describe a compactification of …
The decomposition theorem for smooth projective morphisms $π:\mathcal{X}\rightarrow B$ says that $Rπ_*\mathbb{Q}$ decomposes as $\oplus R^iπ_*\mathbb{Q}[-i]$. We describe simple examples where it is not possible to have such a decomposition compatible with cup-product, even after restriction to Zariski dense open sets of $B$. We prove however that this is always possible for families of $K3$ surfaces (after shrinking the base), and show how this result relates to a result by Beauville and the author on the Chow ring of $K3$ surfaces $S$. We give two proofs of this result, the second one involv…
The Segre Determinant
本文研究Segre行列式，计算其在不同坐标系下的形式，并证明其代表Grassmannian中通用的torus轨道的Chow-Lam形式，同时探讨其在代数视觉和Grassmannian Chow商中的应用。
Rationally Simply Connected Hypersurfaces in Orthogonal Grassmannians
本文研究正交格拉斯曼流形中低次超曲面的有理曲线模空间，并证明满足特定条件的超曲面具有有理数单连通性。
Given a projective variety $X$ of codimension $k+1$ in $\mathbb{P}^n$ the Chow hypersurface $Z_X$ is the hypersurface of the Grassmannian $\operatorname{Gr}(k, n)$ parametrizing projective linear spaces that intersect $X$. We introduce the tropical Chow hypersurface $\operatorname{Trop}(Z_X)$. This object only depends on the tropical variety $\operatorname{Trop}(X)$ and we provide an explicit way to obtain $\operatorname{Trop}(Z_X)$ from $\operatorname{Trop}(X)$. We also give a geometric description of $\operatorname{Trop}(Z_X)$. We conjecture that, as in the classical case, $\operatorname{Tro…
We generalize the notion of coisotropic hypersurfaces to subvarieties of Grassmannians having arbitrary codimension. To every projective variety X, Gel'fand, Kapranov and Zelevinsky associate a series of coisotropic hypersurfaces in different Grassmannians. These include the Chow form and the Hurwitz form of X. Gel'fand, Kapranov and Zelevinsky characterized coisotropic hypersurfaces by a rank one condition on conormal spaces, which we use as the starting point for our generalization. We also study the dual notion of isotropic varieties by imposing rank one conditions on tangent spaces instead…
Deformations of highly symmetric Calabi-Yau Grassmannian hypersurfaces
本文运用算术与Hodge理论技术研究Grassmannians及其商空间中高度对称超曲面实现的Calabi-Yau簇的铅笔，证明其几何性质不同于经典镜像。
Over any complex cubic hypersurface of dimension at least 2, the Chow group of 1-dimensional cycles is spanned by the lines lying on the hypersurface. The smooth case has already been given several other proofs. -- On montre que sur toute hypersurface cubique complexe de dimension au moins 2, le groupe de Chow des cycles de dimension 1 est engendré par les droites. Le cas lisse est un théorème connu. La démonstration ici donnée repose sur un résultat sur les surfaces géométriquement rationnelles sur un corps quelconque (1983), obtenu via la K-théorie algébrique.
Let $Y$ be a smooth dimensionally transverse intersection of the Grassmannian $\hbox{Gr}(2,n)$ with 3 Plücker hyperplanes. We show that $Y$ admits a multiplicative Chow-Künneth decomposition, in the sense of Shen-Vial. As a consequence, a certain tautological subring of the Chow ring of powers of $Y$ injects into cohomology.
Questions on the Chow ring of complete intersections
本文探讨了Calabi-Yau超曲面的Chow环问题，证明了部分结果，并提出了关于MCK分解的猜想。
Motivic Lefschetz theorem for twisted Milnor hypersurfaces
本研究针对Milnor超曲面内扭曲形式的光滑超平面截面，证明其Grothendieck-Chow motive可分解为相关循环代数的Severi-Brauer簇与极大交换子域的motive平移拷贝的直和，证明依赖于Galois作用在等变Chow群上的非平凡性。
We show that points in specific degree 2 hypersurfaces in the Grassmannian $Gr(3, n)$ correspond to generic arrangements of $n$ hyperplanes in $\mathbb{C}^3$ with associated discriminantal arrangement having intersections of multiplicity three in codimension two.
Algebraic hyperbolicity of very general hypersurfaces in homogeneous varieties
本文推广了Coskun、Riedl和Yeong的技巧，得到了有理齐次簇中非常一般超曲面的代数双曲性的几乎最优次数界，并给出了格拉斯曼簇、正交/辛格拉斯曼簇及旗簇等实例。
We give bounds on the order of torsion in the Chow group of zero dimensional cycles for isotropic Grassmannians and Brauer-Severi flag varieties. To do this, we introduce tools to understand the behavior of torsion in Chow groups with coefficients under morphisms of proper varieties
Inspirés par un argument de C. Voisin, nous montrons l'existence d'hypersurfaces quartiques lisses dans ${\bf P}^4_{\mathbb C}$ qui ne sont pas stablement rationnelles, plus précisément dont le groupe de Chow de degré zéro n'est pas universellement égal à $\mathbb Z$. --- There are (many) smooth quartic hypersurfaces in ${\bf P}^4_{\mathbb C}$ which are not stably rational. More precisely, their degree zero Chow group is not universally equal to $\mathbb Z$. The proof uses a variation of a specialisation method due to C. Voisin.
Given a sheaf on a projective space P^n we define a sequence of canonical and easily computable Chow complexes on the Grassmannians of planes in P^n, generalizing the Beilinson monad on P^n. If the sheaf has dimension k, then the Chow form of the associated k-cycle is the determinant of the Chow complex on the Grassmannian of planes of codimension k+1. Using the theory of vector bundles and the canonical nature of the complexes we are able to give explicit determinantal and Pfaffian formulas for resultants in some cases where no polynomial formulas were known. For example, the Horrocks-Mumford…
We prove the vanishing of the subgroup of Bloch's cubical higher Chow groups $CH^2(\text{Spec}(\MF_p),3)$, $p=2,3$, generated by the images of corresponding projective Grassmannian homology groups ${}^PGH_1^2(\MF_p)$ using computer calculations.
We adapt for algebraically closed fields $k$ of characteristic greater than $2$ two results of Voisin, on the decomposition of the diagonal of a smooth cubic hypersurface $X$ of dimension $3$ over $\mathbb C$, namely: the equivalence between Chow-theoretic and cohomological decompositions of the diagonal of those hypersurfaces and the fact that the algebraicity (with $\mathbb Z_2$-coefficients) of the minimal class $θ^4/4!$ of the intermediate jacobian $J(X)$ of $X$ implies the Chow-theoretic decomposition of the diagonal of $X$. Using the second result, the Tate conjecture for divisors on sur…
The minimal degeneration singularities in the affine Grassmannians of simple simply-laced algebraic groups are determined to be either Kleinian singularities of type A, or closures of minimal orbits in nilpotent cones. The singularities for non-simply-laced types are studied by intersection cohomology and equivariant Chow group methods.
We provide several ingredients towards a generalization of the Littlewood-Richardson rule from Chow groups to algebraic cobordism. In particular, we prove a simple product-formula for multiplying classes of smooth Schubert varieties with any Bott-Samelson class in algebraic cobordism of the grassmannian. We also establish some results for generalized Schubert polynomials for hyperbolic formal group laws.
A Quadratically Enriched Count Of Lines In Smooth Del Pezzo Surfaces Of Degree 2 And 4
本文研究光滑del Pezzo曲面度2和4的线计数，探讨Chow-Witt群的Euler类计算，并展示二次计数在Chow群同构下的特性。
Let X be a smooth quasi-projective variety over the algebraic closure of the rational number field. We show that the cycle map of the higher Chow group to Deligne cohomology is injective and the higher Hodge cycles are generated by the image of the cycle map as conjectured by Beilinson and Jannsen, if the cycle map to Deligne cohomology is injective and the Hodge conjecture is true for certain smooth projective varieties over the algebraic closure of the rational number field. We also verify the conjecture on the surjectivity in the case of the complement of a union of general hypersurfaces in…
We study the Chow motive (with rational coefficients) of a hypersurface X in the projective space by using the variety F(X) of l-dimensional planes contained in X. If the degree of X is sufficiently small we show that the primitive part of the motive of X is the tensor product of a direct summand in the motive of a suitable complete intersection in F(X) and the l-th twist Q(-l) of the Lefschetz motive.
On Fano varieties of index one that are linear sections of Grassmannians of lines
本文研究线Grassmannian的一般线性截面Fano簇，证明其双有理等价于由pfaffian行列式定义的超曲面，并给出相应超曲面的有理参数化。
We use a theorem of Chow (1949) on line-preserving bijections of Grassmannians to determine the automorphism group of Grassmann codes. Further, we analyze the automorphisms of the big cell of a Grassmannian and then use it to settle an open question of Beelen et al. (2010) concerning the permutation automorphism groups of affine Grassmann codes. Finally, we prove an analogue of Chow's theorem for the case of Schubert divisors in Grassmannians and then use it to determine the automorphism group of linear codes associated to such Schubert divisors. In the course of this work, we also give an alt…
A remark on the product by the hyperplane class in the Chow ring of some complete intersections
该研究将光滑超曲面上超平面类乘积在同调平凡有理闭链上为零的经典结果，推广至某些完全交的Chow环情形。
The Chow Ring Classes of $\mathrm{PGL}_3$ Orbit Closures in $\mathbb{G}(1, 5)$
本文研究平面圆锥曲线束构成的格拉斯曼簇在 PGL(V) 作用下的八个轨道，通过几何描述、枚举技巧和陈类计算，确定了各轨道闭包在周环中的类及其普吕克嵌入次数。
A congruence is a surface in the Grassmannian ${\rm Gr}(2, 4)$. In this paper, we consider the normalization of congruence of bitangents to a hypersurface in $\mathbb P^3$. We call it the Fano congruence of bitangents. We give a criterion for smoothness of the Fano congruence of bitangents and describe explicitly their degenerations in a general Lefschetz pencil in the space of hypersurfaces in $\mathbb P^3$.
Computing the Chow variety of quadratic space curves
本文研究三维空间直线 Grassmannian 中二次曲面的余迷向超曲面子簇，按 GKZ 理论将其分解为三类轨迹，并计算相应理想。
We define and study the totally nonnegative part of the Chow quotient of the Grassmannian, or more simply the nonnegative configuration space. This space has a natural stratification by positive Chow cells, and we show that nonnegative configuration space is homeomorphic to a polytope as a stratified space. We establish bijections between positive Chow cells and the following sets: (a) regular subdivisions of the hypersimplex into positroid polytopes, (b) the set of cones in the positive tropical Grassmannian, and (c) the set of cones in the positive Dressian. Our work is motivated by connecti…
Let $H \subset {\mathbb P}^n$ be a real-analytic subvariety of codimension one induced by a real-analytic curve in the Grassmannian $G(n+1,n)$. Assuming $H$ has a global defining function, we prove $H$ is Levi-flat, the closure of its smooth points of top dimension is a union of complex hyperplanes, and its singular set is either of dimension $2n-2$ or dimension $2n-4$. If the singular set is of dimension $2n-4$, then we show the hypersurface is algebraic and the Levi-foliation extends to a singular holomorphic foliation of ${\mathbb P}^n$ with a meromorphic (rational of degree 1) first integr…
Consider an external product of a higher cycle and a usual cycle which is algebraically equivalent to zero. Assume there exists an algebraically closed subfield k such that the higher cycle and its ambient variety are defined over k, but the image of the usual cycle by the Abel-Jacobi map is not. Then we prove that the external product is nonzero if the image of the higher cycle by the cycle map to the reduced Deligne cohomology does not vanish. We also give examples of indecomposable higher cycles on even dimensional hypersurfaces of degree at least four in a projective space such that the la…
Positive Polytopes with Few Facets in the Grassmannian
本文研究 Gr(2,4) 与少面简单多面体相交所得正几何的伴随超曲面，证明五面情形唯一，并给出六面情形伴随族维数上界及非唯一仍为正几何的例子。
We construct an explicit regulator map from the weigh n Bloch Higher Chow group complexto the weight n Deligne complex of a regular complex projective algebraic variety X. We define the Arakelovweight n motivic complex as the cone of this map shifted by one. Its last cohomology group is (a version of) the Arakelov Chow group defined by H. Gillet. and C.Soule. We relate the Grassmannian n-logarithms (defined as in [G5]) to geometry of the symmetric space for GL_n(C). For n=2 we recover Lobachevsky's formula for the volume of an ideal geodesic tetrahedron via the dilogarithm. Using the relations…
Let $Y$ be a smooth complete intersection of a quadric and a cubic in $\mathbb{P}^n$, with $n$ even. We show that $Y$ has a multiplicative Chow-Künneth decomposition, in the sense of Shen-Vial. As a consequence, the Chow ring of (powers of) $Y$ displays K3-like behaviour. As a by-product of the argument, we also establish a multiplicative Chow-Künneth decomposition for the resolution of singularities of a general nodal cubic hypersurface of even dimension.
We study the existence of a Chow-theoretic decomposition of the diagonal of a smooth cubic hypersurface, or equivalently, the universal triviality of its ${\rm CH}_0$-group. We prove that for odd dimensional cubic hypersurfaces or for cubic fourfolds, this is equivalent to the existence of a cohomological decomposition of the diagonal, and we translate geometrically this last condition. For cubic threefolds $X$, this turns out to be equivalent to the algebraicity of the minimal class $θ^4/4!$ of the intermediate Jacobian $J(X)$. In dimension $4$, we show that a special cubic fourfold with disc…
Let $X$ be a cubic hypersurface in $\mathbb P^6$ or a hypersurface of degree greater than equal to $7$ in $\mathbb P^5$. In this note we try to understand, for a very general hyperplane section of $X$, the non-injectivity locus of the corresponding push-forward homomorphism at the level of Chow group of certain dimension.
扫码添加微信好友，提出您的宝贵建议 👇
💡 备注请填写：网站反馈
发表机构由 AI 从论文文本抽取，如发现错误，欢迎反馈修正建议。