J. M. Landsberg, L. Manivel
We give an elementary introduction to our papers relating the geometry of rational homogeneous varieties to representation theory. We also describe related work and recent progress.
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
We present a new proof of the classification of complex simple Lie algebras via the projective geometry of homogeneous varieties. Our proof proceeds by constructing homogeneous varieties using the ideals of the secant and tangential varieties of homogeneous varieties already constructed. Our algorithms make no reference to root systems. Our proofs use properties of root systems, but not their classification.
The purpose of this article is to introduce projective geometry over composition algebras : the equivalent of projective spaces and Grassmannians over them are defined. It will follow from this definition that the projective spaces are in correspondance with Jordan algebras and that the points of a projective space correspond to rank one matrices in the Jordan algebra. A second part thus studies properties of rank one matrices. Finally, subvarieties of projective spaces are discussed.
We prove rigidity of various types of holomorphic parabolic geometry on smooth complex projective varieties.
We connect the algebraic geometry and representation theory associated to Freudenthal's magic square. We give unified geometric descriptions of several classes of orbit closures, describing their hyperplane sections and desingularizations, and interpreting them in terms of composition algebras. In particular, we show how a class of invariant quartic polynomials can be viewed as generalizations of the classical discriminant of a cubic polynomial.
These are lecture notes on the rigidity of submanifolds of projective space "resembling" compact Hermitian symmetric spaces in their homogeneous embeddings. Recent results are surveyed, along with their classical predecessors. The notes include an introduction to moving frames in projective geometry, an exposition of the Hwang-Yamaguchi ridgidity theorem and a new variant of the Hwang-Yamaguchi theorem.
We describe the natural geometry of Hilbert schemes of curves in ${\mathbb P}^3$ and, in some cases, in ${\mathbb P}^n$ , $n\geq 4$.
Consider a finite dimensional (generally reducible) polynomial representation ρof GL_n. A projective compactification of GL_n is the closure of ρ(GL_n) in the space of all operators defined up to a factor (this class of spaces can be characterized as equivariant projective normal compactifications of GL_n). We give an expicit description for all projective compactifications. We also construct explicitly (in elementary geometrical terms) a universal object for all projective compactifications of GL_n.
We explore connections between birational anabelian geometry and abstract projective geometry. One of the applications is a proof of a version of the birational section conjecture.
We discuss various phenomena of tangency in projective and convex geometry.
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.
This is an expanded and updated version of a lecture series I gave at Seoul National University in September 1997. It is in some sense an update of the 1979 Griffiths and Harris paper with a similar title. I discuss: Homogeneous varieties, Topology and consequences Projective differential invariants, Varieties with degenerate Gauss images, When can a uniruled variety be smooth?, Dual varieties, Linear systems of bounded and constant rank, Secant and tangential varieties, Systems of quadrics with tangential defects, Recognizing uniruled varieties, Recognizing intersections of quadrics, Recogniz…
Limit geometry of complete projective special real manifolds
本文研究完备射影特殊实流形的极限几何，给出可能的极限几何列表并证明其对称群维数下界，且在正则边界条件下极限几何同构于 $\mathbb{R}_{>0}\ltimes\mathbb{R}^{n-1}$。
We consider the construction of Frobenius manifolds associated to projective special geometry and analyse the dependence on choices involved. In particular, we prove that the underlying F-manifold is canonical. We then apply this construction to integrable systems of Hitchin type.
We study locally free sheaves of rank two on the projective line over the integers, especially indecomposable ones. Subsequently we apply various concepts of Arakelov geometry to these sheaves. We compute for example the arithmetic Chern classes and use the arithmetic Riemann-Roch theorem.
We show that the minimal model program on any smooth projective surface is realized as a variation of the moduli spaces of Bridgeland stable objects in the derived category of coherent sheaves.
This article describes recent applications of algebraic geometry to noncommutative algebra. These techniques have been particularly successful in describing graded algebras of small dimension.
We classify minimal-degree curves in the Hilbert schemes of points on algebraic surfaces. When the algebraic surface is the projective plane, the nef cone and a flip structure of these Hilbert schemes are determined.
Any set of $σ$-Hermitian matrices of size $n \times n$ over a field with involution $σ$ gives rise to a projective line in the sense of ring geometry and a projective space in the sense of matrix geometry. It is shown that the two concepts are based upon the same set of points, up to some notational differences.
To understand the structure of an algebraic variety we often embed it in various projective spaces. This develops the notion of projective geometry which has been an invaluable tool in algebraic geometry. We develop a perfectoid analog of projective geometry, and explore how equipping a perfectoid space with a map to a certain analog of projective space can be a powerful tool to understand its geometric and arithmetic structure. In particular, we show that maps from a perfectoid space $X$ to the perfectoid analog of projective space correspond to line bundles on $X$ together with some extra da…
Bondal-Orlov's reconstruction theorem in noncommutative projective geometry
该研究证明Bondal-Orlov重构定理在非交换射影几何中成立，且非交换射影概形导出范畴间的全忠实正合函子为Fourier-Mukai型，推进了非交换射影几何领域的理论成果。
We describe a close relation between wall crossings in the birational geometry of moduli space of Gieseker stable sheaves $M_H(v)$ on $\bb{P}^2$ and mini-wall crossings in the stability manifold $Stab(D^b(\bb{P}^2))$.
We study the projective geometry of homogeneous varieties $X= G/P\subset P(V)$, where $G$ is a complex simple Lie group, $P$ is a maximal parabolic subgroup and $V$ is the minimal $G$-module associated to $P$. Our study began with the observation that Freudenthal's magic chart could be derived from Zak's theorem on Severi varieties and standard geometric constructions. Our attempt to understand this observation led us to discover further connections between projective geometry and representation theory. Among other things, we calculate the variety of tangent directions to lines on $X$ through …
In this note, we generalize the Proj-construction from usual schemes to blue schemes. This yields the definition of projective space and projective varieties over a blueprint. In particular, it is possible to descend closed subvarieties of a projective space to a canonical F_1-model. We discuss this explicitly in case of the Grassmannian Gr(2,4).
In this article, we will introduce methods of non-standard analysis into projective geometry. Especially, we will analyze the properties of a projective space over a non-Archimedean field. Non-Archimedean fields contain numbers that are smaller than every real number: the so called "infinitesimal" numbers. The theory is well known from non-standard analysis. This enables us to define projective objects that deviate only infinitesimally (in an appropriate metric) from each other. And we show that in most cases operations involving such objects that deviate infinitesimally also experience only i…
By recasting metrical geometry in a purely algebraic setting, both Euclidean and non-Euclidean geometries can be studied over a general field with an arbitrary quadratic form. Both an affine and a projective version of this new theory are introduced here, and the main formulas extend those of rational trigonometry in the plane. This gives a unified, computational model of both spherical and hyperbolic geometries, allows the extension of many results of Euclidean geometry to the relativistic setting, and provides a new metrical approach to algebraic geometry.
Brackets and Projective Geometry in Macaulay2
本文为Macaulay2引入了Brackets包，实现了括号环和Grassmann-Cayley代数，支持特殊线性群不变量计算及射影几何与枚举几何中的伸直算法。
We state the fundamental theorem of projective geometry for semimodules over semirings, which is facilitated by recent work in the study of bases in semimodules defined over semirings. In the process we explore in detail the linear algebra setup over semirings. We also provide more explicit results to understand the implications of our main theorem on maps between tropical lines in the tropical plane. Along with this we also look at geometrical connections to the rich theory of tropical geometry
In the past 15 years a study of ``noncommutative projective geometry'' has flourished. By using and generalizing techniques of commutative projective geometry, one can study certain noncommutative graded rings and obtain results for which no purely algebraic proof is known. For instance, noncommutative graded domains of quadratic growth, or ``noncommutative curves,'' have now been classified by geometric data and these rings must be noetherian. Rings of cubic growth, or ``noncommutative surfaces,'' are not yet classified, but a rich theory is currently forming. In this survey, we describe some…
In the first part of the thesis, we study a classical invariant of projective varieties, the secant defectivity. The second part is devoted to modern algebraic geometry, we study the birational geometry of blow-ups of Grassmannians at points.
We study the birational properties of hypersurfaces in products of projective spaces. In the case of hypersurfaces in P^m x P^n, we describe their nef, movable and effective cones and determine when they are Mori dream spaces. Using these results, we give new simple examples of non-Mori dream spaces and analogues of Mumford's example of a strictly nef line bundle which is not ample.
This is a survey article on mirror symmetry and Fourier-Mukai partners of Calabi-Yau threefolds with Picard number one based on recent works by the authors [HoTa1,2,3,4]. For completeness, mirror symmetry and Fourier-Mukai partners of K3 surfaces are also discussed.
Birational geometry of hypersurfaces in products of weighted projective spaces
本研究扩展了加权射影空间乘积中超曲面的双有理几何理论，确定了Mori梦想空间情形下的所有相关锥、双有理模型及Cox环，并证明了终端Calabi-Yau超曲面的Kawamata-Morrison锥猜想之双有理形式。
Points of bounded height on projective spaces over global function fields via geometry of numbers
本文借助数的几何方法，改编 Schanuel 在数域情形的论证，为全局函数域上射影空间的有界高度点计数结果提供了新证明。
On the axiomatics of projective and affine geometry in terms of line intersection
本文通过显式定义，证明在维数≥3的仿射与射影几何的一阶理论中，以直线相交为原始概念时，不相交关系可正面定义。
We show that the cyclic and epicyclic categories which play a key role in the encoding of cyclic homology and the lambda operations, are obtained from projective geometry in characteristic one over the infinite semifield F of "max-plus integers". Finite dimensional vector spaces are replaced by modules defined by restriction of scalars from the one-dimensional free module, using the Frobenius endomorphisms of F. The associated projective spaces are finite and provide a mathematically consistent interpretation of J. Tits' original idea of a geometry over the absolute point. The self-duality of …
We extend the classical Schubert calculus of enumerative geometry for the Grassmann variety of lines in projective space from the complex realm to the real. Specifically, given any collection of Schubert conditions on lines in projective space which generically determine a finite number of lines, we show there exist real generic conditions determining the expected number of real lines. Our main tool is an explicit description of rational equivalences which also constitutes a novel determination of the Chow rings of these Grassmann varieties.
In this paper, we give three bases for the cohomology groups of the Hilbert scheme of two points on projective space. Then, we use these bases to compute all effective and nef cones of higher codimensional cycles on the Hilbert scheme. Next, we compute the class in one of these bases of the Chern classes of tautological bundles coming from line bundles. Finally, we provide an application of these results to the degrees of secant varieties of complete intersections.
We use the cut and paste relation $[Y^{[2]}] = [Y][\mathbb{P}^m] + \mathbb{L}^2 [F(Y)]$ in $K_0(\text{Var}_k)$ of Galkin--Shinder for cubic hypersurfaces arising from projective geometry to characterize cubic hypersurfaces of sufficiently high dimension under certain numerical or genericity conditions. Removing the conditions involving the middle Betti number from the numerical conditions used extends the possible $Y$ to cubic hypersurfaces, complete intersections of two quadric hypersurfaces, or complete intersections of two quartic hypersurfaces. The same method also gives a family of other …
By exploring the consequences of the triviality of the monodromy group for a class of surfaces of which the mixed Hodge structure is pure, we extend results of Miyanishi and Sugie, Dimca, Zaidenberg and Kaliman.
Semi-Projective Representations and Twisted Representation Groups
本文将Schur的投射表示群概念推广到代数闭域上的半投射表示，并给出计算机算法，可生成给定有限群在复数域平凡或共轭作用下的所有扭表示群。
We apply the combinatorial theory of spherical varieties to characterize the momentum polytopes of polarized projective spherical varieties. This enables us to derive a classification of these varieties, without specifying the open orbit, as well as a classification of all Fano spherical varieties. In the setting of multiplicity free compact and connected Hamiltonian manifolds, we obtain a necessary and sufficient condition involving momentum polytopes for such manifolds to be Kähler and classify the invariant compatible complex structures of a given Kähler multiplicity free compact and connec…
We interpret the Landau-Ginzburg potentials associated to Gross-Hacking-Keel-Kontsevich's partial compactifications of cluster varieties as F-polynomials of projective representations of Jacobian algebras. Along the way, we show that both the projective and the injective representations of Jacobi-finite quivers with potential are well-behaved under Derksen-Weyman-Zelevinsky's mutations of representations.
On the continuity of Weil-Petersson volumes of the moduli space weighted points on the projective line
本文证明了射影直线上加权点的模空间中，从Fano几何趋近Calabi-Yau几何时，Weil-Petersson体积连续，且局部化计算的CM体积收敛到McMullen的几何体积。
Line bundles of rational degree are defined using Perfectoid spaces, and their co-homology computed via standard Čech complex along with Kunneth formula. A new concept of `braided dimension' is introduced, which helps convert the curse of infinite dimensionality into a boon, which is then used to do Bezout type computations, define euler characters, describe ampleness and link integer partitions with geometry. This new concept of 'Braided dimension' gives a space within a space within a space an infinite tower of spaces, all intricately braided into each other. Finally, the concept of Blow Up …
Studying the mirror symmetry of a Calabi-Yau threefold $X$ of the Reye congruence in $\mP^4$, we conjecture that $X$ has a non-trivial Fourier-Mukai partner $Y$. We construct $Y$ as the double cover of a determinantal quintic in $\mP^4$ branched over a curve. We also calculate BPS numbers of both $X$ and $Y$ (and also a related Calabi-Yau complete intersection $\tilde X_0$) using mirror symmetry.
We study tetrahedral quartics in projective space. We address their projective geometry, Neron-Severi lattice and automorphism group.
We construct explicit examples of elementary extremal contractions, both birational and of fiber type, from smooth projective n-dimensional varieties, n\geq 4, onto smooth projective varieties, arising from classical projective geometry and defined over sufficiently small fields, not necessarily algebraically closed. The examples considered come from particular special homaloidal and subhomaloidal linear systems, which usually are degenerations of general phenomena classically investigated by Bordiga, Severi, Todd, Room, Fano, Semple and Tyrrell and more recently by Ein and Shepherd-Barron. Th…
In this paper we combine methods from projective geometry, Klein's model, and Clifford algebra. We develop a Clifford algebra whose Pin group is a double cover of the group of regular projective transformations. The Clifford algebra we use is constructed as homogeneous model for the five-dimensional real projective space $P^5(\mathbb{R})$ where Klein's quadric $M_2^4$ defines the quadratic form. We discuss all entities that can be represented naturally in this homogeneous Clifford algebra model. Projective automorphisms of Klein's quadric induce projective transformations of $P^3(\mathbb{R})$ …
We study finite dimensional representations of the projective modular group. Various explicit dimension formulas are given.
The Gale transform, an involution on sets of points in projective space, appears in a multitude of guises, in subjects as diverse as optimization, coding theory, theta-functions, and recently in our proof that certain general sets of points fail to satisfy the minimal free resolution conjecture. In this paper we reexamine the Gale transform in the light of modern algebraic geometry. We give a more general definition, in the context of finite (locally) Gorenstein subschemes. We put in modern form a number of the more remarkable examples discovered in the past, and we add new constructions and c…
Using toric geometry we prove a Bézout type theorem for weighted projective spaces.
We study geometric properties of certain obstructed equisingular families of projective hypersurfaces with emphasis on smoothness, reducibility, being reduced, and having expected dimension. In the case of minimal obstructness, we give a detailed description of such families corresponding to quasihomogeneous singularities. Next we study the behavior of these properties with respect to stable equivalence of singularities. We show that under certain conditions, stabilization of singularities ensures the existence of a reduced component of expected dimension. For minimally obstructed families the…
In this paper we study the topology of the spaces Hol(M,P{n},k) of (basepoint preserving) holomorphic maps of a given degree k from a Riemann surface M of genus g>0 into the n-th complex projective space P{n}, n>0. Using symmetric products of the surface as well as the geometry of the associated Abel-Jacobi maps, we construct an explicit topological compactification of Hol(M,P{n},k) which allow us to construct spectral sequences with known E_1-terms that can be used to determine the homology groups of Hol(M,P{n},k). Complete calculations are given for the elliptic case (g=1). Complete rational…
Generalized weight properties of resultants and discriminants, and applications to projective enumerative geometry
本文研究了结果式和判别式的广义权重性质，并将其应用于射影计数几何中的计数问题。
The Hilbert scheme of point modules was introduced by Artin-Tate-Van den Bergh to study non-commutative graded algebras. The key tool is the construction of a map from the algebra to a twisted ring on this Hilbert scheme. In this paper, we study moduli stacks of more general "fat" point modules, and show that there is a similar map to a twisted ring associated to the stack. This is used to provide a sufficient criterion for a non-commutative projective surface to be birationally PI. It is hoped that such a criterion will be useful in understanding Mike Artin's conjecture on the birational clas…
Gaussian likelihood geometry of projective varieties
本文推广了高斯极大似然度，探讨了齐次多项式在射影簇上的极大似然度，证明其等于有理函数临界点计数，并给出基于拓扑欧拉示性数、对偶簇和陈类的几何刻画。
In this paper we consider orthogonal geometry of the free $\protect\ZZ$-module $K_0(\protect\PP_n)$ with respect to the non-symmetric unimodular bilinear form $$χ(E,F)=\sum (-1)^ν\dim\ext^ν(E,F). $$ We calculate the isometry group of this form and describe invariants of its natural action on $K_0(\protect\PP_n)$. Also we consider some general constructions with non-symmetric unimodular forms. In particular, we discuss orthogonal decomposition of such forms and the action of the braid group on a set of semiorthonormal bases. We formulate a list of natural arithmetical conjectures about semiorth…
Given a projective manifold $X$ equipped with an ample line bundle $L$, we show how to embed certain torus-invariant domains $D \subseteq\mathbb{C}^n$ into $X$ so that the Euclidean Kähler form on $D$ extends to a Kähler form on X lying in the first Chern class of $L$. This is done using Okounkov bodies $Δ(L)$, and the image of $D$ under the standard moment map will approximate $Δ(L)$. This means that the volume of $D$ can be made to approximate the Kähler volume of $X$ arbitrarily well. As a special case we can let $D$ be an ellipsoid. We also have similar results when $L$ is just big.
Projective Representations I. Projective lines over rings
本文研究含单位元的环上的射影直线在域上的射影空间中的表示，基于双模构造，并探讨远距离点与非远距离点与互补子空间的对应关系。
We discuss two results about projective representations of fundamental groups of quasiprojective varieties. The first is a realization result which, under a nonresonance assumption, allows to realize such representations as monodromy representations of flat projective logarithmic connections. The second is a lifting result: any representation as above, after restriction to a Zariski open set and finite pull-back, can be lifted to a linear representation.
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。