共查询到20条相似文献,搜索用时 31 毫秒
1.
We show that if a smooth projective 4-fold M contains an ample divisor A which is P
1-bundle π :A→S over a smooth projective surface S, π is extended to a P
2-bundle π :S→S, unless $A$ is isomorphic to P
2×P
1.
Received: 28 September 1998 / Revised version: 16 August 1999 相似文献
2.
Ravi Vakil 《Inventiones Mathematicae》2006,164(3):569-590
We consider the question: “How bad can the deformation space of an object be?” The answer seems to be: “Unless there is some a priori reason otherwise, the deformation space may be as bad as possible.” We show this for a number of important moduli spaces. More precisely, every singularity of finite type over ? (up to smooth parameters) appears on: the Hilbert scheme of curves in projective space; and the moduli spaces of smooth projective general-type surfaces (or higher-dimensional varieties), plane curves with nodes and cusps, stable sheaves, isolated threefold singularities, and more. The objects themselves are not pathological, and are in fact as nice as can be: the curves are smooth, the surfaces are automorphism-free and have very ample canonical bundle, the stable sheaves are torsion-free of rank 1, the singularities are normal and Cohen-Macaulay, etc. This justifies Mumford’s philosophy that even moduli spaces of well-behaved objects should be arbitrarily bad unless there is an a priori reason otherwise. Thus one can construct a smooth curve in projective space whose deformation space has any given number of components, each with any given singularity type, with any given non-reduced behavior. Similarly one can give a surface over $\mathbb{F}_{p}We consider the question: “How bad can the deformation space of an object be?” The answer seems to be: “Unless there is some
a priori reason otherwise, the deformation space may be as bad as possible.” We show this for a number of important moduli
spaces.
More precisely, every singularity of finite type over ℤ (up to smooth parameters) appears on: the Hilbert scheme of curves
in projective space; and the moduli spaces of smooth projective general-type surfaces (or higher-dimensional varieties), plane
curves with nodes and cusps, stable sheaves, isolated threefold singularities, and more. The objects themselves are not pathological,
and are in fact as nice as can be: the curves are smooth, the surfaces are automorphism-free and have very ample canonical
bundle, the stable sheaves are torsion-free of rank 1, the singularities are normal and Cohen-Macaulay, etc. This justifies
Mumford’s philosophy that even moduli spaces of well-behaved objects should be arbitrarily bad unless there is an a priori
reason otherwise.
Thus one can construct a smooth curve in projective space whose deformation space has any given number of components, each
with any given singularity type, with any given non-reduced behavior. Similarly one can give a surface over
that lifts to ℤ/p7 but not ℤ/p8. (Of course the results hold in the holomorphic category as well.)
It is usually difficult to compute deformation spaces directly from obstruction theories. We circumvent this by relating them
to more tractable deformation spaces via smooth morphisms. The essential starting point is Mn?v’s universality theorem.
Mathematics Subject Classification (2000) 14B12, 14C05, 14J10, 14H50, 14B07, 14N20, 14D22, 14B05 相似文献
3.
Marco Andreatta 《manuscripta mathematica》2003,110(4):505-512
We consider some conditions under which a smooth projective variety X is actually the projective space. We also extend to the case of positive characteristic some results in the theory of vector
bundle adjunction. We use methods and techniques of the so called Mori theory, in particular the study of rational curves
on projective manifolds.
Received: 16 May 2002 / Revised version: 18 November 2002 Published online: 3 March 2003
Mathematics Subject Classification (2000): 14E30, 14J40, 14J45 相似文献
4.
We develop the fundamentals of hereditary noetherian categories with Serre duality over an arbitrary field k, where the category of coherent sheaves over a smooth projective curve over k serves as the prime example and others are coming from the representation theory of finite dimensional algebras. The proper
way to view such a category is to think of coherent sheaves on a possibly non-commutative smooth projective curve. We define
for each such category notions like function field and Euler characteristic, determine its Auslander-Reiten components and
study stable and semistable bundles for an appropriate notion of degree. We provide a complete classification of hereditary
noetherian categories for the case of positive Euler characteristic by relating these to finite dimensional representations
of (locally bounded) hereditary k-algebras whose underlying valued quiver admits a positive additive function.
Dedicated to Otto Kerner on the occasion of his 60th birthday 相似文献
5.
We show that each Jordan homomorphism R → R′ of rings gives rise to a harmonic mapping of one connected component of the projective line over R into the projective line over R′. If there is more than one connected component then this mapping can be extended in various ways to a harmonic mapping which
is defined on the entire projective line over R.
Received December 7, 2001; in revised form April 28, 2002
Published online January 7, 2003 相似文献
6.
Let G be a reductive group over an algebraically closed field k. Consider the moduli space of stable principal G-bundles on a smooth projective curve C over k. We give necessary and sufficient conditions for the existence of Poincaré bundles over open subsets of this moduli space,
and compute the orders of the corresponding obstruction classes. This generalizes the previous results of Newstead, Ramanan
and Balaji–Biswas–Nagaraj–Newstead to all reductive groups, to all topological types of bundles, and also to all characteristics. 相似文献
7.
Let k be an algebraically closed field and X a smooth projective variety defined over k. Let EG be a principal G–bundle over X, where G is an algebraic group defined over k, with the property that for every smooth curve C in X the restriction of EG to C is the trivial G–bundle. We prove that the principal G–bundle EG over X is trivial. We also give examples of nontrivial principal bundle over a quasi-projective variety Y whose restriction to every smooth curve in Y is trivial. 相似文献
8.
S. Subramanian 《Archiv der Mathematik》2007,89(1):68-72
We show that a principal G-bundle on a smooth projective curve over a finite field is strongly semistable if and only if it is defined by a representation
of the fundamental group scheme of the curve into G.
Received: 24 April 2006 相似文献
9.
Given a smooth projective curve X of genus at least 2 over a number field k, Grothendieck’s Section Conjecture predicts that the canonical projection from the étale fundamental group of X onto the absolute Galois group of k has a section if and only if the curve has a rational point. We show that there exist curves where the above map has a section
over each completion of k but not over k. In the appendix Victor Flynn gives explicit examples in genus 2. Our result is a consequence of a more general investigation
of the existence of sections for the projection of the étale fundamental group ‘with abelianized geometric part’ onto the
Galois group. We also point out the relation to the elementary obstruction of Colliot-Thélène and Sansuc.
This article has an appendix by E. V. Flynn. 相似文献
10.
The existence and uniqueness of H-N reduction for the Higgs principal bundles over nonsingular projective variety is shown.
We also extend the notion of H-N reduction for (Γ,G)-bundles and ramifiedG-bundles over a smooth curve. 相似文献
11.
Hans Havlicek Andrzej Matraś Mark Pankov 《Abhandlungen aus dem Mathematischen Seminar der Universit?t Hamburg》2011,81(2):237-249
Given the algebra T of ternions (upper triangular 2×2 matrices) over a commutative field F we consider as set of points of a projective line over T the set of all free cyclic submodules of T
2. This set of points can be represented as a set of planes in the projective space over F
6. We exhibit this model, its adjacency relation, and its automorphic collineations. Despite the fact that T admits an F-linear antiautomorphism, the plane model of our projective line does not admit any duality. 相似文献
12.
Richard Pink 《manuscripta mathematica》2000,102(1):1-24
Let X be an irreducible smooth projective curve over an algebraically closed field of characteristic p>0. Let ? be either a finite field of characteristic p or a local field of residue characteristic p. Let F be a constructible étale sheaf of $\BF$-vector spaces on X. Suppose that there exists a finite Galois covering π:Y→X such that the generic monodromy of π*
F is pro-p and Y is ordinary. Under these assumptions we derive an explicit formula for the Euler–Poincaré characteristic χ(X,F) in terms of easy local and global numerical invariants, much like the formula of Grothendieck–Ogg–Shafarevich in the case
of different characteristic. Although the ordinariness assumption imposes severe restrictions on the local ramification of
the covering π, it is satisfied in interesting cases such as Drinfeld
modular curves.
Received: 7 December 1999 / Revised version: 28 January 2000 相似文献
13.
Richard Bödi 《Archiv der Mathematik》1999,73(1):73-80
We prove that the only compact projective Hughes planes which are smooth projective planes are the classical planes over the complex numbers \Bbb C \Bbb C , the quaternions \Bbb H \Bbb H , and the Caley numbers \Bbb O \Bbb O . As a by-product this shows that an 8-dimensional smooth projective plane which admits a collineation group of dimension d 3 17d \geq 17 is isomorphic to the quaternion projective plane P 2\Bbb H {\cal P _2\Bbb H }. For topological compact projective planes this is true if d 3 19d \geq 19, and this bound is sharp. 相似文献
14.
Diego Izquierdo 《Comptes Rendus Mathematique》2017,355(3):268-271
Let K be the function field of a smooth projective curve X over a higher-dimensional local field k. We define Tate–Shafarevich groups of a commutative group scheme via cohomology classes locally trivial at each completion of K coming from a closed point of X. In this note, we state and sketch the proof of an arithmetic duality theorem for Tate–Shafarevich groups of groups of multiplicative type over K (and more generally of some two-term complexes of tori over K). 相似文献
15.
Let X be a smooth complex projective variety with Neron–Severi group isomorphic to ℤ, and D an irreducible divisor with normal crossing singularities. Assume 1<r≤ 3. We prove that if π1(X) doesn't have irreducible PU(r) representations, then π1(X- D) doesn't have irreducible U(r) representations. The proof uses the non-existence of certain stable parabolic bundles. We also obtain a similar result for
GL(2) when D is smooth.
Received: 20 December 1999 / Revised version: 7 May 2000 相似文献
16.
Xiaotao Sun 《Inventiones Mathematicae》2008,173(2):427-447
Let X be a smooth projective variety of dimension n over an algebraically closed field k with char(k)=p>0 and F:X→X
1 be the relative Frobenius morphism. For any vector bundle W on X, we prove that instability of F
*
W is bounded by instability of W⊗Tℓ(Ω1
X
) (0≤ℓ≤n(p-1)) (Corollary 4.9). When X is a smooth projective curve of genus g≥2, it implies F
*
W being stable whenever W is stable.
Dedicated to Professor Zhexian Wan on the occasion of his 80th birthday. 相似文献
17.
We study projective resolutions for quasi-finite comodules over a semiperfect coalgera. We show that the number of the indecomposable projective comodules in the i-th term of the projective resolution for these comodules is finite, which gives an invariant, the so-called Betti numbers. We study the behavior of these invariants under duality and localization. 相似文献
18.
Mohamed Saïdi 《manuscripta mathematica》2005,118(4):425-441
In this paper we prove that complete families of smooth and projective curves of genus g≥2 in characteristic p>0 with a constant geometric fundamental group are isotrivial. 相似文献
19.
We define a quasi–projective reduction of a complex algebraic variety X to be a regular map from X to a quasi–projective variety that is universal with respect to regular maps from X to quasi–projective varieties. A toric quasi–projective reduction is the analogous notion in the category of toric varieties.
For a given toric variety X we first construct a toric quasi–projective reduction. Then we show that X has a quasi–projective reduction if and only if its toric quasi–projective reduction is surjective. We apply this result
to characterize when the action of a subtorus on a quasi–projective toric variety admits a categorical quotient in the category
of quasi–projective varieties.
Received October 29, 1998; in final form December 28, 1998 相似文献
20.
Ji-hong SU & Yi-cai ZHAO LMAM Department of Mathematics Jinan University Guangzhou China 《中国科学A辑(英文版)》2007,50(4):495-502
Let X be a smooth projective variety of dimension 2k-1 (k≥3) over the complex number field. Assume that fR: X→Y is a small contraction such that every irreducible component Ei of the exceptional locus of fR is a smooth subvariety of dimension k. It is shown that each Ei is isomorphic to the k-dimensional projective space Pk, the k-dimensional hyperquadric surface Qk in Pk 1, or a linear Pk-1-bundle over a smooth curve. 相似文献