SEARCH · Searcharxiv

Results for “alg-geom”

Search indexed arXiv papers on artificial intelligence, large language models, computer vision and robotics. Read source abstracts and follow links to arXiv.

Quote a phrase for an exact phrase match. Source license links do not imply unrestricted reuse.

At least 343 records · Page 19Linked to original sources

arXiv1997

Kawachi's invariant for normal surface singularities

We study a useful numerical invariant of normal surface singularities, introduced recently by T. Kawachi. Using this invariant, we give a quick proof of the (well-known) fact that all log-canonical surface singularities are either elliptic Gorenstein or rational (without assuming a priori that they are Q-Gorenstein). The main result is a criterion (stated in terms of Kawachi's invariant) regarding global generation of adjoint linear systems on normal surfaces with boundary. Such results can be used in proving effective estimates for global generation on singular threefolds. The theorem of Ein-Lazarsfeld and Kawamata, which says that the minimal center of log-canonical singularities is always normal, explains why the results proved here are relevant in that situation.

alg-geom↗

arXiv1997

Very ampleness of adjoint linear systems on smooth surfaces with boundary

Let M be a Q-divisor on a smooth surface over C. In this paper we give criteria for very ampleness of the adjoint of the round-up of M. (Similar results for global generation were given by Ein and Lazarsfeld and used in their proof of Fujita's Conjecture in dimension 3.) In the last section we discuss an example which suggests that this kind of criteria might also be useful in the study of linear systems on surfaces.

alg-geom↗

arXiv1998

Wall-Crossing functors and D-modules

We study Translation functors and Wall-Crossing functors on infinite dimensional representations of a complex semisimple Lie algebra using D-modules. This functorial machinery is then used to prove the Endomorphism-theorem and the Structure-theorem, two important results established earlier by W. Soergel in a totally different way. Other applications to the category O of Bernstein-Gelfand-Gelfand are given, and some conjectural relationships between Koszul duality, Verdier duality and convolution functors are discussed. A geometric interpretation of tilting modules is given.

alg-geom↗

arXiv1997

A generalization of curve genus for ample vector bundles, II

Let $X$ be a compact complex manifold of dimension $n\ge 2$ and $\ce$ an ample vector bundle of rank $r<n$ on $X$. As the continuation of Part I, we further study the properties of $g(X,\ce)$ that is an invariant for pairs $(X,\ce)$ and is equal to curve genus when $r=n-1$. Main results are the classifications of $(X,\ce)$ with $g(X,\ce)=2$ (resp. 3) when $\ce$ has a regular section (resp. $\ce$ is ample and spanned) and $1<r<n-1$.

alg-geom↗

arXiv1997

On Moduli of G-bundles over Curves for exceptional G

Let $G$ be a simple and simply connected complex Lie group, ${\goth{g}}$ its Lie algebra. I remove the restriction ``$G$ is of classical type or $G_2$'' made on $G$ in the papers of Beauville, Laszlo and myself [L-S] and [B-L-S] on the moduli of principal G-bundles over a curve. As I will just "patch" the missing technical points, this note should be seen as an appendix to the above cited papers.

alg-geom↗

arXiv1997

On the width of lattice-free simplices

Among integral polytopes (vertices with integral coordinates), lattice-free polytopes - intersecting the lattice ONLY at their vertices- are of particular interestin combinatorics and geometry of numbers. A natural question is to measure their "width" (with respect to the integral lattice).There were no known examples of lattice-free polytopes with width bigger than 2 .We prove the following Theorem : Given any positive number $α$ strictly inferior to $1/e$, for d large enough there exists a lattice-free simplex of dimension d and width superior to $αd$.

alg-geom↗

arXiv1997

On fibre space structures of a projective irreducible symplectic manifold

In this note, we investigate fibre space structures of a projective irreducible symplectic manifold. We prove that an 2n-dimensional projective irreducible symplectic manifold admits only an n-dimensional fibration over a Fano variety which has only Q-factorial log-terminal singularities and whose Picard number is one. Moreover we prove that a general fibre is an abelian variety up to finite unramified cover, especially, a general fibre is an abelian surface for 4-fold.

alg-geom↗

arXiv1997

An algebraic cell decomposition of the nonnegative part of a flag variety

We study the nonnegative part B_{\ge 0} of the flag variety of a reductive algebraic group G, as defined by Lusztig. Using positivity properties of the canonical basis it is shown that B_{\ge 0} has an algebraic cell decomposition indexed by pairs w\le w' of the Weyl group. This result was conjectired by Lusztig in [Lu; Progress in Math 123].

alg-geom↗

arXiv1997

On higher order embeddings of Fano threefolds by the anticanonical linear system

The map given by the anticanonical bundle of a Fano manifold is investigated with respect to a number of natural notions of higher order embeddings of projective manifolds. This is of importance in the understanding of higher order embeddings of the special varieties of adjunction theory, which are usually fibered by special Fano manifolds. An analysis is carried out of the higher order embeddings of the special varieties of adjunction theory that arise in the study of the first and second reductions. Special attention is given to determining the order of the anticanonical embeddings of the three dimensional Fano manifolds which have been classified by Iskovskih, Mori, and Mukai and also of the Fano complete intersections in $\pn N$.

alg-geom↗

arXiv1997

Resonant Hypergeometric Systems and Mirror Symmetry

The Gamma-series of Gel'fand-Kapranov-Zelevinsky are adapted so that they give solutions for certain resonant systems of GKZ hypergeometric differential equations. For this some complex parameters in the Gamma-series are replaced by nilpotent elements from a ring $R_{A,T}$. The adapted Gamma-series is a function $Ψ$ with values in the finite dimensional vector space $R_{A,T}\otimes C$. Applications of these results in the context of toric Mirror Symmetry are described. Building on work of Batyrev we show that the relative cohomology module of a certain hypersurface in a torus is a GKZ hypergeometric $D$-module which over an appropriate domain is isomorphic to the trivial $D$-module $R_{A,T}\otimes O_T$, where $O_T$ is the sheaf of holomorphic functions on this domain. The isomorphism is explicitly given by adapted Gamma-series. As a result one finds the periods of a holomorphic differential form of degree $d$ on a $d$-dimensional Calabi-Yau manifold, needed for the B-model side input to Mirror Symmetry. Relating our work with that of Batyrev and Borisov we interpret the ring $\cR_{\sA,\gT}$ as the cohomology ring of a toric variety and a certain principal ideal in it as a subring of the Chow ring of a Calabi-Yau complete intersection. This interpretation takes place on the A-model side of Mirror Symmetry.

alg-geom↗

arXiv2010

Some fundamental problems on real-analytic sets

The category of real-analytic sets and real-analytic maps is the most important category in application. However, in spite of efforts by F. Bruhat, H. Cartan, H. Whitney et al., the basic theory of real-analytic category does not yet seem to be well-developed. In this article I would like to point out several basic problems.

alg-geom↗

arXiv2008

Semiinfinite Flags. II. Local and Global Intersection Cohomology of Quasimaps' Spaces

For a simple algebraic group $G$ we study the space $Q$ of Quasimaps from the projective line $C$ to the flag variety of $G$. We prove that the global Intersection Cohomology of $Q$ carries a natural pure Tate Hodge structure, and compute its generating function. We define an action of the Langlands dual Lie algebra $g^L$ on this cohomology. We present a new geometric construction of the universal enveloping algebra $U(n^L_+)$ of the nilpotent subalgebra of $g^L$. It is realized in the Ext-groups of certain perverse sheaves on Quasimaps' spaces, and it is equipped with a canonical basis numbered by the irreducible components of certain algebraic cycles, isomorphic to the intersections of semiinfinite orbits in the affine Grassmannian of $G$. We compute the stalks of the IC-sheaves on the Schubert strata closures in $Q$. They carry a natural pure Tate Hodge structure, and their generating functions are given by the generic affine Kazhdan-Lusztig polynomials. In the Appendix we prove that Kontsevich's space of stable maps provides a natural resolution of $Q$.

alg-geom↗

arXiv1997

First quantum correction for the moduli space of stable bundles over a Riemann surface

We compute some Gromov-Witten invariants of the moduli space of odd degree rank two stable vector bundles over a Riemann surface of any genus. Next we find the first correction term for the quantum product of this moduli space and hence get the two leading terms of the relations satisfied by the natural generators of its quantum cohomology. Finally, we use this information to get a full description of the quantum cohomology of the moduli space when the genus of the Riemann surface is 3.

alg-geom↗

arXiv1997

Boundary Manifolds of Line Arrangements

In this paper we describe the complement of real line arrangements in the complex plane in terms of the boundary three-manifold of the line arrangement. We show that the boundary manifold of any line arrangement is a graph manifold with Seifert fibered vertex manifolds, and depends only on the incidence graph of the arrangement. When the line arrangement is defined over the real numbers, we show that the homotopy type of the complement is determined by the incidence graph together with orderings on the edges emanating from each vertex.

alg-geom↗

arXiv1997

Stringy Hodge numbers and Virasoro algebra

Let $X$ be an arbitrary smooth $n$-dimensional projective variety. It was discovered by Libgober and Wood that the product of the Chern classes $c_1(X)c_{n-1}(X)$ depends only on the Hodge numbers of $X$. This result has been used by Eguchi, Jinzenji and Xiong in their approach to the quantum cohomology of $X$ via a representation of the Virasoro algebra with the central charge $c_n(X)$. In this paper we define for singular varieties $X$ a rational number $c_{st}^{1,n-1}(X)$ which is a stringy version of the number $c_1c_{n-1}$ for smooth $n$-folds. We show that the number $c_{st}^{1,n-1}(X)$ can be expressed in the same way using the stringy Hodge numbers of $X$. Our results provides an evidence for the existence of an approach to quantum cohomology of singular varieties $X$ via a representation of the Virasoro algebra whose central charge is the rational number $e_{st}(X)$ which equals the stringy Euler number of $X$.

alg-geom↗

arXiv1997

A geometric approach to the fundamental lemma for unitary groups

We consider from a geometric point of view the conjectural fundamental lemma of Langlands and Shelstad for unitary groups over a local field of positive characteristic. We introduce projective algebraic varieties over the finite residue field $k$ and interpret the conjecture in this case as a remarkable identity between the number of $k$-rational points of them. We prove the corresponding identity for the numbers of $k_f$-rational points, for any extension of even degree $f$ of $k$. The proof uses the local intersection theory on a regular surface and Deligne's theory of intersection multiplicities with weights. We also discuss a possible descent argument that uses $\ell$-adic cohomology to treat extensions of odd degree as well.

alg-geom↗

arXiv1999

Equations of the moduli of pointed curves in the infinite Grassmannian

The main result of this paper is the explicit computation of the equations defining the moduli space of triples $(C,p,z)$ (where $C$ is an integral and complete algebraic curve, $p$ a smooth rational point and $z$ a formal trivialization around $p$) in the infinite Grassmannian of $k((t))$. This is achieved by introducing infinite Grassmannians, tau and Baker-Ahkiezer functions algebraically and by proving an Addition Formula for tau functions.

alg-geom↗

arXiv1997

Cellular Resolutions of Monomial Modules

We construct a canonical free resolution for arbitrary monomial modules and lattice ideals. This includes monomial ideals and defining ideals of toric varieties, and it generalizes our joint results with Irena Peeva for generic ideals.

alg-geom↗