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1.
Summary We are interested in counting integer and rational points in affine algebraic varieties, also under congruence conditions. We introduce the notions of a strongly Hardy-Littlewood variety and a relatively Hardy-Littlewood variety, in terms of counting rational points satisfying congruence conditions. The definition of a strongly Hardy-Littlewood variety is given in such a way that varieties for which the Hardy-Littlewood circle method is applicable are strongly Hardy-Littlewood.We prove that certain affine homogeneous spaces of semisimple groups are strongly Hardy-Littlewood varieties. Moreover, we prove that many homogeneous spaces are relatively Hardy-Littlewood, but not strongly Hardy-Littlewood. This yields a new class of varieties for with the asymptotic density of integer points can be computed in terms of a product of local densities.Oblatum 15-IX-1993 & 31-I-1994 相似文献
2.
F. Bihan 《Commentarii Mathematici Helvetici》2003,78(2):227-244
Let be a nonsingular real algebraic surface of degree m in the complex projective space and its real point set in . In the spirit of the sixteenth Hilbert's problem, one can ask for each degree m about the maximal possible value of the Betti number (i=0 or 1). We show that is asymptotically equivalent to for some real number and prove inequalities and .
Received: April 26, 2000 相似文献
3.
4.
Peter Bürgisser Felipe Cucker Martin Lotz 《Foundations of Computational Mathematics》2005,5(4):351-387
In [8] counting complexity classes #PR and #PC in the Blum-Shub-Smale (BSS) setting of
computations over the real and complex numbers, respectively,
were introduced. One of the main results of [8] is that the problem
to compute the Euler characteristic of a semialgebraic set
is complete in the class FPR#PR.
In this paper, we prove that the corresponding result is true
over C, namely that the computation of the Euler characteristic
of an affine or projective complex variety is complete
in the class FPC#PC. We also obtain a corresponding
completeness result for the Turing model. 相似文献
5.
We exhibit some new techniques to study volumes of tubes about algebraic varieties in complex projective spaces. We prove
the existence of relations between volumes and Intersection Theory in the presence of singularities. In particular, we can
exhibit an average Bezout Equality for equidimensional varieties. We also state an upper bound for the volume of a tube about
a projective variety. As a main outcome, we prove an upper bound estimate for the volume of the intersection of a tube with
an equidimensional projective algebraic variety. We apply these techniques to exhibit upper bounds for the probability distribution
of the generalized condition number of singular complex matrices. 相似文献
6.
The action of an affine algebraic group G on an algebraic variety V can be
differentiated to a representation of the Lie algebra L(G) of G by derivations on the sheaf
of regular functions on V . Conversely, if one has a finite-dimensional Lie algebra L and a
homomorphism ρ : L → DerK(K[U]) for an affine algebraic variety U, one may wonder whether it comes from an algebraic group action on U or on a variety
V containing U as an open subset. In this paper, we prove two results on this integration problem. First, if L acts faithfully
and locally finitely on K[U], then it can be embedded in L(G), for some affine algebraic group G acting on U, in such a way
that the representation of L(G) corresponding to that action restricts to ρ on L. In the second theorem, we assume from the
start that L = L(G) for some connected affine algebraic group G and show that some technical but necessary conditions on ρ
allow us to integrate ρ to an action of G on an algebraic variety V containing U as an open dense subset. In the interesting
cases where L is nilpotent or semisimple, there is a natural choice for G, and our technical conditions take a more appealing
form. 相似文献
7.
Johan Huisman 《Discrete and Computational Geometry》2005,33(1):157-163
The set of all unordered real line arrangements of given degree in the
real projective plane is known to have a natural semialgebraic
structure. The nonreduced arrangements are singular points of this
structure. We show that the set of all unordered real line
arrangements of given degree also has a natural structure of a smooth
compact connected affine real algebraic variety. In fact, as such, it
is isomorphic to a real projective space. As a consequence, we get a
projectively linear structure on the set of all real line arrangements
of given degree. We also show that the universal family of unordered
real line arrangements of given degree is not algebraic. 相似文献
8.
Nero Budur 《Advances in Mathematics》2009,221(1):217-250
The space of unitary local systems of rank one on the complement of an arbitrary divisor in a complex projective algebraic variety can be described in terms of parabolic line bundles. We show that multiplier ideals provide natural stratifications of this space. We prove a structure theorem for these stratifications in terms of complex tori and convex rational polytopes, generalizing to the quasi-projective case results of Green-Lazarsfeld and Simpson. As an application we show the polynomial periodicity of Hodge numbers hq,0 of congruence covers in any dimension, generalizing results of E. Hironaka and Sakuma. We extend the structure theorem and polynomial periodicity to the setting of cohomology of unitary local systems. In particular, we obtain a generalization of the polynomial periodicity of Betti numbers of unbranched congruence covers due to Sarnak-Adams. We derive a geometric characterization of finite abelian covers, which recovers the classic one and the one of Pardini. We use this, for example, to prove a conjecture of Libgober about Hodge numbers of abelian covers. 相似文献
9.
M. Boij J. Migliore R.M. Miró-Roig U. Nagel 《Journal of Pure and Applied Algebra》2019,223(4):1456-1471
Musta?? has given a conjecture for the graded Betti numbers in the minimal free resolution of the ideal of a general set of points on an irreducible projective algebraic variety. For surfaces in this conjecture has been proven for points on quadric surfaces and on general cubic surfaces. In the latter case, Gorenstein liaison was the main tool. Here we prove the conjecture for general quartic surfaces. Gorenstein liaison continues to be a central tool, but to prove the existence of our links we make use of certain dimension computations. We also discuss the higher degree case, but now the dimension count does not force the existence of our links. 相似文献
10.
Motivated by finding an effective way to compute the algebraic complexity of the nearest point problem for algebraic models, we introduce an efficient method for detecting the limit points of the stratified Morse trajectories in a small perturbation of any polynomial function on a complex affine variety. We compute the multiplicities of these limit points in terms of vanishing cycles. In the case of functions with only isolated stratified singularities, we express the local multiplicities in terms of polar intersection numbers. 相似文献
11.
JongHae Keum 《Topology》2006,45(5):919-927
A fake projective plane is a compact complex surface (a compact complex manifold of dimension 2) with the same Betti numbers as the complex projective plane, but not isomorphic to the complex projective plane. As was shown by Mumford, there exists at least one such surface.In this paper we prove the existence of a fake projective plane which is birational to a cyclic cover of degree 7 of a Dolgachev surface. 相似文献
12.
We continue the study of counting complexity begun in [13], [14], [15] by proving upper and lower bounds on the complexity
of computing the Hilbert polynomial of a homogeneous ideal. We show that the problem of computing the Hilbert polynomial of
a smooth equidimensional complex projective variety can be reduced in polynomial time to the problem of counting the number
of complex common zeros of a finite set of multivariate polynomials. The reduction is based on a new formula for the coefficients
of the Hilbert polynomial of a smooth variety. Moreover, we prove that the more general problem of computing the Hilbert polynomial
of a homogeneous ideal is polynomial space hard. This implies polynomial space lower bounds for both the problems of computing
the rank and the Euler characteristic of cohomology groups of coherent sheaves on projective space, improving the #P-lower
bound in Bach [1]. 相似文献
13.
D. Kotschick 《Advances in Mathematics》2012,229(2):1300-1312
We prove that a rational linear combination of Chern numbers is an oriented diffeomorphism invariant of smooth complex projective varieties if and only if it is a linear combination of the Euler and Pontryagin numbers. In dimension at least three we prove that only multiples of the top Chern number, which is the Euler characteristic, are invariant under diffeomorphisms that are not necessarily orientation-preserving. These results solve a long-standing problem of Hirzebruch's. We also determine the linear combinations of Chern numbers that can be bounded in terms of Betti numbers. 相似文献
14.
Bertrand To?n 《Selecta Mathematica, New Series》2005,12(1):39-134
The purpose of this work is to introduce a notion of affine stacks, which is a homotopy version of the notion of affine schemes, and to give several applications in the context of algebraic
topology and algebraic geometry.
As a first application we show how affine stacks can be used in order to give a new point of view (and new proofs) on rational
and p-adic homotopy theory. This gives a first solution to A. Grothendieck’s schematization problem described in [18].
We also use affine stacks in order to introduce a notion of schematic homotopy types. We show that schematic homotopy types give a second solution to the schematization problem, which also allows us to go beyond
rational and p-adic homotopy theory for spaces with arbitrary fundamental groups. The notion of schematic homotopy types is also used in
order to construct various homotopy types of algebraic varieties corresponding to various co-homology theories (Betti, de
Rham, l-adic, ...), extending the well known constructions of the various fundamental groups.
Finally, just as algebraic stacks are obtained by gluing affine schemes we define
$$ \infty $$-geometric stacks as a certain gluing of affine stacks. Examples of
$$ \infty $$-geometric stacks in the context of algebraic topology (moduli spaces of dga structures up to quasi-isomorphisms)
and Hodge theory (non-abelian periods) are given. 相似文献
15.
Bertrand Toën 《Selecta Mathematica, New Series》2006,12(1):39-134
The purpose of this work is to introduce a notion of affine stacks, which is a homotopy version of the notion of affine schemes, and to give several applications in the context of algebraic
topology and algebraic geometry.
As a first application we show how affine stacks can be used in order to give a new point of view (and new proofs) on rational
and p-adic homotopy theory. This gives a first solution to A. Grothendieck’s schematization problem described in [18].
We also use affine stacks in order to introduce a notion of schematic homotopy types. We show that schematic homotopy types give a second solution to the schematization problem, which also allows us to go beyond
rational and p-adic homotopy theory for spaces with arbitrary fundamental groups. The notion of schematic homotopy types is also used in
order to construct various homotopy types of algebraic varieties corresponding to various co-homology theories (Betti, de
Rham, l-adic, ...), extending the well known constructions of the various fundamental groups.
Finally, just as algebraic stacks are obtained by gluing affine schemes we define
$$ \infty $$-geometric stacks as a certain gluing of affine stacks. Examples of
$$ \infty $$-geometric stacks in the context of algebraic topology (moduli spaces of dga structures up to quasi-isomorphisms)
and Hodge theory (non-abelian periods) are given. 相似文献
16.
We prove that any affine, resp. polarized projective, spherical variety admits a flat degeneration to an affine, resp. polarized projective, toric variety. Motivated by mirror symmetry, we give conditions for the limit toric variety to be a Gorenstein Fano, and provide many examples. We also provide an explanation for the limits as boundary points of the moduli space of stable pairs whose existence is predicted by the Minimal Model Program. 相似文献
17.
We prove that any affine, resp. polarized projective, spherical variety admits a flat degeneration to an affine, resp. polarized
projective, toric variety. Motivated by mirror symmetry, we give conditions for the limit toric variety to be a Gorenstein
Fano, and provide many examples. We also provide an explanation for the limits as boundary points of the moduli space of stable
pairs whose existence is predicted by the Minimal Model Program. 相似文献
18.
《Mathematische Nachrichten》2017,290(17-18):2992-2994
We give an upper bound for the number of cusps of a plane affine or projective curve via its first Betti number. 相似文献
19.
Daniel Greb 《Advances in Mathematics》2010,224(2):401-431
Given an action of a complex reductive Lie group G on a normal variety X, we show that every analytically Zariski-open subset of X admitting an analytic Hilbert quotient with projective quotient space is given as the set of semistable points with respect to some G-linearised Weil divisor on X. Applying this result to Hamiltonian actions on algebraic varieties, we prove that semistability with respect to a momentum map is equivalent to GIT-semistability in the sense of Mumford and Hausen. It follows that the number of compact momentum map quotients of a given algebraic Hamiltonian G-variety is finite. As further corollary we derive a projectivity criterion for varieties with compact Kähler quotient. 相似文献
20.
Real plane algebraic curves 总被引:1,自引:0,他引:1
Maria Jesus de la Puente 《Expositiones Mathematicae》2002,20(4):291-314
We study real algebraic plane curves, at an elementary level, using as little algebra as possible. Both cases, affine and projective, are addressed. A real curve is infinite, finite or empty according to the fact that a minimal polynomial for the curve is indefinite, semi-definite nondefinite or definite. We present a discussion about isolated points. By means of the P operator, these points can be easily identified for curves defined by minimal polynomials of order bigger than one. We also discuss the conditions that a curve must satisfy in order to have a minimal polynomial. Finally, we list the most relevant topological properties of affine and projective, complex and real plane algebraic curves. 相似文献