Interpolation Problems for Vector Bundles on Algebraic Curves

In this paper we study meromorphic maps between vector bundles on a Riemann surface. We are mainly interested in stable vector bundles. For a huge number of numerical data we prove the existence of a meromorphic map between two vector bundles with a prescribed number of zeroes and a prescribed number of poles.     

The Speciality Lemma, Rank 2 Bundles and Gherardelli-type Theorems for Surfaces in P 4

In the present paper we give a very short and easy proof of the speciality lemma for codimension 2 subvarieties, even those that are reducible or non-reduced, in P ⁿ . Furthermore we give cohomological conditions that force a subcanonical surface in P ⁴ to be a complete intersection and a rank 2 bundle to split, which generalize the classical First Theorem of Gherardelli.     

Nodal Curves with General Moduli on K3 Surfaces

We investigate the modular properties of nodal curves on a low genus K3 surface. We prove that a general genus g curve C is the normalization of a δ-nodal curve X sitting on a primitively polarized K3 surface S of degree 2p ? 2, for 2 ≤ g = p ? δ < p ≤ 11. The proof is based on a local deformation-theoretic analysis of the map from the stack of pairs (S, X) to the moduli stack of curves ? _g that associates to X the isomorphism class [C] of its normalization.     

Rank 2 Ample Vector Bundles on Some Smooth Rational Surfaces

Several classification results for ample vector bundles of rank 2 on Hirzebruch surfaces, and on Del Pezzo surfaces, are obtained. In particular, we classify rank 2 ample vector bundles with _c2 less than seven on Hirzebruch surfaces, and with _c2 less than four on Del Pezzo surfaces.     

Boundaries of Flat Compact Surfaces in 3-Space

This paper deals with the problem `which knots or links in3-space bound flat (immersed) compact surfaces?' In aforthcoming paper by the author, it is proven that any simple closedspace curve can be deformed until it bounds a flat orientable compact(Seifert) surface. The main results of this paper are that there existknots that do not bound any flat compact surfaces. The lower bound oftotal curvature of a knot bounding an orientable nonnegatively curvedcompact surface can, for varying knot types, be arbitrarily much greaterthan the infimum of curvature needed for the knot to have its knot type.The number of 3-singular points (points of zero curvatureor if not then of zero torsion) on the boundary of a flat immersedcompact surface is greater than or equal to twice the absolute value ofthe Euler characteristic of the surface. A set of necessary and, in aweakened sense, sufficient conditions for a knot or link to be what wecall a generic boundary of a flat immersed compact surface withoutplanar regions is given.     

Moduli of Vector Bundles on Curves in Positive Characteristics

Let X be a projective curve of genus 2 over an algebraically closed field of characteristic 2. The Frobenius map on X induces a rational map on the moduli scheme of rank-2 bundles. We show that up to isomorphism, there is only one (up to tensoring by an order two line bundle) semi-stable vector bundle of rank 2 (with determinant equal to a theta characteristic) whose Frobenius pull-back is not semi-stable. The indeterminacy of the Frobenius map at this point can be resolved by introducing Higgs bundles.     

Some results on special stable vector bundles of rank 3 on algebraic curves

The authors discuss the existence and classification of stable vector bundles of rank 3, with 2 3 or 4 linearly independent holomorphic sections. The sets of all such bundles are denoted by ω3^2,d and w3 respectively. Our argument leads to sufficient and necessary conditions for the existence of both kinds of bundles. The conclusion is very interesting because of its contradiction to the conjectured dimension formula of stable bundles. Finally, we give a preliminary classification of ω3^2,4 and a complete discussion on the structure of ω3^3,2/3g＋2.     

The Limiting Shape of the Generic Initial System of a Complete Intersection

Consider a complete intersection I of type (d ₁,…, d _r ) in a polynomial ring over a field of characteristic 0. We study the graded system of ideals {gin(I ⁿ )} _n obtained by taking the reverse lexicographic generic initial ideals of the powers of I and describe its asymptotic behavior. This behavior is nicely captured by the limiting shape which is shown to depend only on the type of the complete intersection.     

Factoriality of Certain Threefolds Complete Intersection in P 5 with Ordinary Double Points

Abstract

Let V ? P ⁵ be a reduced and irreducible threefold of degree s, complete intersection on a smooth hypersurface of degree t, with s > t ² ? t. In this paper, we prove that if the singular locus of V consists of δ < 3s/8t ordinary double points, then any projective surface contained in V is a complete intersection on V. In particular, V is Q-factorial.     

Complete Sets of Relations in the Cohomology Rings of Moduli Spaces of Holomorphic Bundles and Parabolic Bundles Over a Riemann Surface

The cohomology ring of the moduli space of stable holomorphicvector bundles of rank n and degree d over a Riemann surfaceof genus g > 1 has a standard set of generators when n andd are coprime. When n = 2 the relations between these generatorsare well understood, and in particular a conjecture of Mumford,that a certain set of relations is a complete set, is knownto be true. In this article generalisations are given of Mumford'srelations to the cases when n > 2 and also when the bundlesare parabolic bundles, and these are shown to form completesets of relations. 2000 Mathematics Subject Classification 14H60.     

Moduli Spaces of Coherent Systems of Small Slope on Algebraic Curves

Let C be an algebraic curve of genus g ≥ 2. A coherent system on C consists of a pair (E, V), where E is an algebraic vector bundle over C of rank n and degree d and V is a subspace of dimension k of the space of sections of E. The stability of the coherent system depends on a parameter α. We study the geometry of the moduli space of coherent systems for 0 < d ≤ 2n. We show that these spaces are irreducible whenever they are nonempty and obtain necessary and sufficient conditions for nonemptiness.     

Monomial Ideals and the Gorenstein Liaison Class of a Complete Intersection

In an earlier work, the authors described a mechanism for lifting monomial ideals to reduced unions of linear varieties. When the monomial ideal is Cohen–Macaulay (including Artinian), the corresponding union of linear varieties is arithmetically Cohen–Macaulay. The first main result of this paper is that if the monomial ideal is Artinian then the corresponding union is in the Gorenstein linkage class of a complete intersection (glicci). This technique has some interesting consequences. For instance, given any (d + 1)-times differentiable O-sequence H, there is a nondegenerate arithmetically Cohen–Macaulay reduced union of linear varieties with Hilbert function H which is glicci. In other words, any Hilbert function that occurs for arithmetically Cohen–Macaulay schemes in fact occurs among the glicci schemes. This is not true for licci schemes. Modifying our technique, the second main result is that any Cohen–Macaulay Borel-fixed monomial ideal is glicci. As a consequence, all arithmetically Cohen–Macaulay subschemes of projective space are glicci up to flat deformation.     

Generalized Adjunction of k-Very Ample Vector Bundles on Algebraic Surfaces

We study the k-very ampleness of the adjoint bundle K _S+det E associated to a k-very ample vector bundle E on an algebraic surface S. We extend the results of Beltrametti and Sommese to the vector bundle case and give the classification of the pairs (S, E) such that the preservation of k-very ampleness fails.     

Reducible Families of Curves with Ordinary Multiple Points on Surfaces in IP3 C

For a Coxeter system (G, S) the multi-parametric alternating subalgebra H ⁺(G) of the Hecke algebra and the alternating subgroup ?⁺(G) of the braid group are defined. Two presentations for H ⁺(G) and ?⁺(G) are given; one generalizes the Bourbaki presentation for the alternating subgroups of Coxeter groups, another one uses generators related to edges of the Coxeter graph.     

A Note on Smooth Surfaces in P4

We exhibit a simpler and quick proof of a theorem proved by G. Ellingsrud and Ch. Peskine, which states that there are only a finite number of components in the Hilbert scheme parametrizing smooth surfaces in P⁴ not of general type.     

Vector bundles on a product of real cubic curves

In this paper we give a characterization of then-tuples (C ₁,...,C _n ) of nonsingular projective real cubic curves such that every topological complex vector bundle onC ₁×...×C _n admits an algebraic structure. The results are very explicit and can be expressed in an especially simple form for cubies defined over the rationals.The second author was supported by an NSF grant.     

一类分形曲面的精细计盒维数

研究一类分形曲面的精细计盒维数,得到了星积分形曲面与其生成元的精细计盒维数的关系.