H. Lange, P. E. Newstead
In this paper, we remove a technical hypothesis from a previous paper. This allows us to construct stable bundles of rank 2 with Clifford index significantly smaller than the classical Clifford index on suitably chosen curves of any genus $\ge13$.
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In this paper, we determine bundles which compute the higher Clifford indices for trigonal curves.
Clifford indices for semistable vector bundles on a smooth projective curve of genus at least 4 were defined in a previous paper of the authors. The present paper studies bundles which compute these Clifford indices. We show that under certain conditions on the curve all such bundles and their Serre duals are generated.
Clifford indices of vector bundles on algebraic curves were introduced in a previous paper of the authors. In this paper we study bundles of rank 2 which compute these Clifford indices. This is of particular interest in the light of recently discovered counterexamples to a conjecture of Mercat.
For smooth projective curves the Clifford index is an important invariant which provides a bound for the dimension of the space of sections of a line bundle. This is the first step in distinguishing curves of the same genus. In this paper we generalise this to introduce Clifford indices for semistable vector bundles on curves. We study these invariants, giving some basic properties and carrying out some computations for small ranks and for general and some special curves. For curves whose classical Clifford index is two, we compute all values of our new Clifford indices.
Two definitions of the Clifford index for vector bundles on a smooth projective curve $C$ of genus $g\ge4$ were introduced in a previous paper by the authors. In another paper the authors obtained results on one of these indices for bundles of rank 3. In this paper we extend these results in the case where $C$ has classical Clifford index 3. In particular we prove Mercat's conjecture for bundles of rank 3 for $g \leq 8$ and $g \geq 13$ when $C$ has classical Clifford index 3. We obtain complete results in the case of genus 7.
Let $X$ be a smooth projective K3 surface over complex numbers and $C$ be an ample curve on $X$. In this paper we will study the semistability of the Lazarsfeld-Mukai bundle $E_{C, A}$ associated to a line bundle $A$ ion $C$ such that $|A|$ is a pencil on $C$ and computes the Clifford index of $C$. We give a necessary and sufficient condition for $E_{C, A}$ being semistable.
Clifford indices for semistable vector bundles on a smooth projective curve of genus at least 4 were defined in previous papers of the authors. In the present paper, we establish lower bounds for the Clifford indices for rank 3 bundles. As a consequence we show that, on smooth plane curves of degree at least 10, there exist non-generated bundles of rank 3 computing one of the Clifford indices.
Let $C$ be a smooth irreducible projective algebraic curve defined over the complex numbers. The notion of the Clifford index of $C$ was extended a few years ago to semistable bundles of any rank. Recent work has been focussed mainly on the rank-2 Clifford index, although interesting results have also been obtained for the case of rank 3. In this paper we extend this work, obtaining improved lower bounds for the rank-3 Clifford index. This allows the first computations of the rank-3 index in non-trivial cases and examples for which the rank-3 index is greater than the rank-2 index.
In this paper we obtain bounds on $h^0(E)$ where $E$ is a semistable bundle of rank 3 over a smooth irreducible projective curve $X$ of genus $g \geq 2$ defined over an algebraically closed field of characteristic 0. These bounds are expressed in terms of the degrees of stability $s_1(E), s_2(E)$. We show also that in some cases the bounds are best possible. These results extend recent work of J. Cilleruelo and I. Sols for bundles of rank 2.
Let $(X,H)$ be a polarized K3 surface with $\mathrm{Pic}(X) = \mathbb Z H$, and let $C\in |H|$ be a smooth curve of genus $g$. We give an upper bound on the dimension of global sections of a semistable vector bundle on $C$. This allows us to compute the higher rank Clifford indices of $C$ with high genus. In particular, when $g\geq r^2\geq 4$, the rank $r$ Clifford index of $C$ can be computed by the restriction of Lazarsfeld-Mukai bundles on $X$ corresponding to line bundles on the curve $C$. This is a generalization of the result by Green and Lazarsfeld for curves on K3 surfaces to higher ra…
We establish for smooth projective real curves the equivalent of the classical Clifford inequality known for complex curves. We also study the cases when equality holds.
We discuss natural connections between three objects: quadratic forms with values in line bundles, conic bundles and quaternion orders. We use the even Clifford algebra, and the Brauer-Severi Variety, and other constructions to give natural bijections between these objects under appropriate hypothesis. We then restrict to a surface base and we express the second Chern class of the order in terms $K^3$ and other invariants of the corresponding conic bundle. We find the conic bundles corresponding to minimal del Pezzo quaterion orders and we discuss problems concerning their moduli.
It has been a long-standing problem to find an adequate definition of a Clifford index for higher rank vector bundles on curves, which should capture the complexity of the curve in its moduli space. An interesting proposal in rank 2 has been put forward by Mercat, who conjectured that the second Clifford index of a curve should be equal to its classical Clifford index, defined in terms of gonality. Using moduli of sheaves on generic K3 surfaces, we prove Mercat's conjecture for generic curves of every genus. Furthermore, for odd g we identify an effective divisor in the moduli space M_g along …
Let $X$ be a semistable curve and $L$ a line bundle whose multidegree is uniform, i.e., in the range between those of the structure sheaf and the dualizing sheaf of $X$. We establish an upper bound for $h^0(X,L)$, which generalizes the classic Clifford inequality for smooth curves. The bound depends on the total degree of $L$ and connectivity properties of the dual graph of $X$. It is sharp, in the sense that on any semistable curve there exist line bundles with uniform multidegree that achieve the bound.
On relative Ulrich bundles and generalized Clifford algebras
本文研究了相对乌里奇束与广义克利夫代数之间的 functorial 等价关系，并探讨了相对超曲面的乌里奇-野性性质及同调障碍的解决方法。
Linear series on curves: stability and Clifford index
本文研究光滑复曲线上线性系的稳定性概念，揭示对偶跨度丛稳定性与线性稳定性的联系，得到与 Clifford 指数相关的稳定性条件，并构造出积分斜率 3 的稳定向量丛及其 theta 因子。
This is the author's 2008 thesis from the University of Chicago. We generalize the notion of the Clifford index to an arbitrary very ample line bundle and show how it determines when a curve and its various secant varieties have determinantal equations. In particular we establish the scheme theoretic version of a conjecture of Eisenbud-Koh-Stillman. If $L_1$ and $L_2$ are line bundles of degree at least $2g+1+k$, then ${\rm Sec}^j(C)$ is determinantally presented for $j \leq k$ and for $j \leq k+1$ if $L_1 \neq L_2$. We also give a geometric characterization of the standard Clifford index in t…
We give a sharp lower bound for the selfintersection of a nef line bundle $L$ on an irregular variety $X$ in terms of its continuous global sections and the Albanese dimension of $X$, which we call the Generalized Clifford-Severi inequality. We also extend the result to nef vector bundles and give a slope inequality for fibred irregular varieties. As a byproduct we obtain a lower bound for the volume of irregular varieties; when $X$ is of maximal Albanese dimension the bound is $Vol(X) \geq 2 n! χω_X$ and it is sharp.
In this article, we provide an overview of a one-to-one correspondence between representations of the generalized Clifford algebra $C_f$ of a ternary cubic form $f$ and certain vector bundles (called Ulrich bundles) on a cubic surface $X$. We study general properties of Ulrich bundles, and using a recent classification of Casanellas and Hartshorne, deduce the existence of irreducible representations of $C_f$ of every possible dimension.
We extend the results of Pareschi on the constancy of the gonality and Clifford index of smooth curves in a complete linear system on Del Pezzo surfaces of degrees $\geq 2$ to the case of Del Pezzo surfaces of degree 1, where we explicitly classify the cases where the gonality and Clifford index are not constant. We also classify all cases of exceptional curves on Del Pezzo surfaces, which turn out to be the smooth plane curves and some other cases with Clifford dimension 3. Moreover, the property of being exceptional holds for all curves in the complete linear system. Furthermore, we relate t…
In this paper we construct new examples of stable bundles of rank 2 of small degree with 4 sections on a smooth irreducible curve of maximal Clifford index. The corresponding Brill-Noether loci have negative expected dimension of arbitrarily large absolute value.
Clifford representatives via the uniform algebraic rank
本文在有限图上引入一致代数秩，证明其介于Caporaso代数秩与Baker-Norine组合秩之间，建立相应Riemann-Roch定理，并借此证明克利福德代表总存在且对一大类图给出显式构造。
Higher rank Clifford's theorem on the smooth quadric
本文将 Clifford 定理推广至 P^1×P^1 上的高秩斜率半稳定层，依据秩与第一陈类给出 h^0(E) 的精确上界，并刻画等号情形及相关扩张的半稳定性。
We give two generalizations of the Clifford theorem to algebraic surfaces. As an application, we obtain some bounds for the number of moduli of surfaces of general type.
Classifying binary quadratic forms using Clifford invariants
本文利用Clifford不变量对二元二次型进行分类，并推广高斯组合与二次代数Picard群的关系。
We investigate stratified-algebraic vector bundles on a real algebraic variety X. A stratification of X is a finite collection of pairwise disjoint, Zariski locally closed subvarieties whose union is X. A topological vector bundle on X is called a stratified-algebraic vector bundle if, roughly speaking, there exists a stratification of X such that the restriction of the bundle to each stratum is an algebraic vector bundle. In particular, every algebraic vector bundle on X is stratified-algebraic. It turns out that stratified-algebraic vector bundles have many surprising properties, which disti…
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 $P^3(\mathbb{R})$ …
Let S be a K3 surface and assume for simplicity that it does not contain any (-2)-curve. Using coherent systems, we express every non-simple Lazarsfeld-Mukai bundle on S as an extension of two sheaves of some special type, that we refer to as generalized Lazarsfeld-Mukai bundles. This has interesting consequences concerning the Brill-Noether theory of curves C lying on S. From now on, let g denote the genus of C and A be a complete linear series of type g^r_d on C such that d<= g-1 and the corresponding Brill-Noether number is negative. First, we focus on the cases where A computes the Cliffor…
Let $(X, H)$ be a polarized smooth projective algebraic surface and $E$ is globally generated, stable vector bundle on $X$. Then the Syzygy bundle $M_E$ associated to it is defined as the kernel bundle corresponding to the evaluation map. In this article we will study the stability property of $M_E$ with respect to $H$.
We prove an existence result for stable vector bundles with arbitrary rank on an algebraic surface, and determine the birational structure of certain moduli space of stable bundles on a rational ruled surface.
Linear stability and rank two Clifford indices of algebraic curves with applications
本文证明计算秩二 Clifford 指标的向量丛线性半稳定，给出线性稳定条件，并应用于相干系统，验证 Butler 猜想在特定范围内成立。
In this note, we study monodromies of algebraic connections on the trivial vector bundle. We prove that on a smooth complex affine curve, any monodromy arises as the underlying local system of an algebraic connection on the trivial bundle. We give a generalization of this for rank one monodromies in higher dimension.
We extend the notion of the spectrum of semistable rank two bundles (initially constructed on ${\bf P}^3$ by Barth and Elencwajg) to a certain class of compact complex algebraic threefolds.
Green's conjecture is proved for smooth curves C lying on a rational surface S with an anticanonical pencil, under some mild hypotheses on the line bundle L defined by C. Constancy of Clifford dimension, Clifford index and gonality of curves in the linear system |L| is also obtained.
We give a theorem of Leray-Hirsch type for Chow groups and use it to study the Hogde and Grothendieck's standard conjectures for algebraic fiber bundles of Leray-Hirsch type. Morevoer, the Hodge conjecture for product varieties will also be considered.
Cohomology of Moduli Stacks of Principal $\Bbb C^*$-Bundles Over Nodal Algebraic Curves
研究结点曲线上主$\Bbb C^*$-丛模栈，用陈类确定其有理上同调代数。
We construct smooth rational real algebraic varieties of every dimension $\ge$ 4 which admit infinitely many pairwise non-isomorphic real forms.
Higher dimensional Clifford-Severi equalities
本文刻画了高维Clifford-Severi不等式取等号时的簇与线丛条件，给出了等式成立的完整分类。
Given a very ample line bundle L on a projective variety X, the syzygy bundle M_L associated to L is the kernel of the evaluation map on sections of L. Our main result is that if X is a smooth projective surface defined over an algebraically closed field, then M_L is slope-stable for any sufficiently positive L.
Let X be a geometrically connected smooth projective curve of genus one, defined over the field of real numbers, such that X does not have any real points. We classify the isomorphism classes of all stable real algebraic vector bundles over X.
This is a survey on the topic explained in the title, for the proceedings on the K-theory 1997 summer institute in Seattle.
This article is based in part on lecture notes prepared for the summer school "The Geometry, Topology and Physics of Moduli Spaces of Higgs Bundles" at the Institute for Mathematical Sciences at the National University of Singapore in July of 2014. The aim is to provide a brief introduction to algebraic stacks, and then to give several constructions of the moduli stack of Higgs bundles on algebraic curves. The first construction is via a "bootstrap" method from the algebraic stack of vector bundles on an algebraic curve. This construction is motivated in part by Nitsure's GIT construction of a…
Lee and Pandharipande studied a "double point" algebraic cobordism theory of varieties equipped with vector bundles, and speculated that some features of that story might extend to the case of varieties with principal G-bundles. This note shows that this expectation holds rationally, and more generally after inverting the torsion index of the group, for reductive G. We show that (after inverting the torsion index) the full theory for bundles on varieties is an extension of scalars of standard algebraic cobordism, that the theory for a point is dual to Omega^*(BG), and describe how the theory f…
Let C be a 2-connected Gorenstein curve either reduced or contained in a smooth algebraic surface and let S be a subcanonical cluster (i.e. a 0-dim scheme such that the space H^0(C, I_S K_C) contains a generically invertible section). Under some general assumptions on S or C we show that h^0(C, I_S K_C) <= p_a(C) - deg (S)/2 and if equality holds then either S is trivial, or C is honestly hyperelliptic or 3-disconnected. As a corollary we give a generalization of Clifford's theorem for reduced curves.
Algebraic parabolic bundles on smooth projective curves over algebraically closed field of positive characteristic is defined. It is shown that the category of algebraic parabolic bundles is equivalent to the category of orbifold bundles defined in \cite{KP}. Tensor, dual, pullback and pushforward operations are also defined for parabolic bundles.
In this paper we consider various notions of positivity for distributions on complex projective manifolds. We start by analyzing distributions having big slope with respect to curve classes, obtaining characterizations of generic projective space bundles in terms of movable curve classes. We then apply this result to investigate algebraicity of leaves of foliations, providing a lower bound for the algebraic rank of a foliation in terms of invariants measuring positivity. We classify foliations attaining this bound, and describe those whose algebraic rank slightly exceeds this bound.
We study the spaces of stable real and quaternionic vector bundles on a real algebraic curve. The basic relationship is established with unitary representations of an extension Z/2 by the fundamental group. By comparison with the space of real or quaternionic connections, some of the basic topological invariants of these spaces are calculated.
Let $L$ be a globally generated line bundle over a smooth irreducible complex projective surface $X$. The syzygy bundle $M_{L}$ is the kernel of the evaluation map $H^0(L)\otimes\mathcal O_X\to L$. We prove the $L$-stability of $M_L$ for Hirzebruch surfaces, del Pezzo surfaces and Enriques surfaces. The $(-K_X)$-stability of syzygy bundles $M_L$ over del Pezzo surfaces is also obtained.
Vector bundles in positive characteristics have a tendency to be destabilized after pulling back by the Frobenius morphism. In this paper, we closely examine vector bundles over curves that are, in an appropriate sense, maximally destabilized by the Frobenius morphism. Then we prove that such bundles of rank 2 exist over any curve in characteristic 3, and are unique up to twisting by a line bundle. We also give an application of such bundles to the study of ample vector bundles, which is valid in all characteristics.
Algebraic algorithms for vector bundles over curves
本文提出用函数域极大序上的格对表示曲线上的向量丛，并给出计算行列式、同态核像、同构、上同调、扩张及不可分解直和分解的多项式时间代数算法，同时实现为SageMath包并应用于椭圆曲线向量丛分类与代数几何码构造。
Let $X$ be a normal, connected and projective variety over an algebraically closed field $k$. It is known that a vector bundle $V$ on $X$ is essentially finite if and only if it is trivialized by a proper surjective morphism $f:Y\to X$. In this paper we introduce a different approach to this problem which allows to extend the results to normal, connected and strongly pseudo-proper algebraic stack of finite type over an arbitrary field $k$.
Characterising semi-Clifford gates using algebraic sets
本文研究容错量子计算中的 Clifford 层级门，证明任意素数维度下最多两个 qudit 的第三层门均为半 Clifford 门，并利用代数几何构造概形，证明其模 $d$ 约化后的代数集有理点相同，推广了早期结果。
Given a nondegenerate ternary form $f=f(x_1,x_2,x_3)$ of degree 4 over an algebraically closed field of characteristic zero, we use the geometry of K3 surfaces and van den Bergh's correspondence between representations of the generalized Clifford algebra $C_f$ associated to $f$ and Ulrich bundles on the surface $X_f:=\{w^{4}=f(x_1,x_2,x_3)\} \subseteq \mathbb{P}^3$ to construct a positive-dimensional family of irreducible representations of $C_f.$ The main part of our construction, which is of independent interest, uses recent work of Aprodu-Farkas on Green's Conjecture together with a result …
We show that holomorphic bundles on O(-k) for k > 0 are algebraic. We also show holomorphic bundles on O(-1) are trivial outside the zero section.
We construct examples of non-isomorphic algebraic vector bundles on the punctured affine space with isomorphic pullbacks to the smooth quadric.
The double point relation defines a natural theory of algebraic cobordism for bundles on varieties. We construct a simple basis (over the rationals) of the corresponding cobordism groups over Spec(C) for all dimensions of varieties and ranks of bundles. The basis consists of split bundles over products of projective spaces. Moreover, we prove the full theory for bundles on varieties is an extension of scalars of standard algebraic cobordism.
We determined the Picard group of the moduli of rank two stable sheaves on an arbitrary algebraic surface up to finite index
In this note we study two features of submanifolds (subvarieties) with ample normal bundles in a compact Kähler manifold X. First, we study how algebraic X can be, i.e. we investigate the algebraic dimension. Second, we study curves with ample normal bundles in case X is projective. We prove in various cases that the class of the curve is in the interior of the Mori cone.
We propose an algebraic Riemann-Roch formula for moving flat bundles on contant families in characteristic zero with values in the ring of algebraic differential characters. This formula lifts Grothendieck-Riemann-Roch formula in the Chow groups and Bismut-Lott formula in the odd dimensional Betti cohomology with $\C/\Z$ coefficients over the field of complex numbers.
We prove that the forgetful morphism from the moduli space of orthogonal bundles to the moduli space of vector bundles over a smooth curve is an embedding. Our proof relies on an explicit description of a set of generators for the invariants on the representation space of a quiver under the action of a product of classical groups.
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