Multigraded regularity and the Koszul property

We give a criterion for the section ring of an ample line bundle to be Koszul in terms of multigraded regularity. We discuss applications to adjoint bundles on toric varieties as well as to polytopal semigroup rings.     

A note on general blow-ups of abelian surfaces

In the paper we give numerical conditions for a line bundle on a general blow-up of abelian surface to be k-very ample.     

Higher Embeddings of General Blowups of Ruled Surfaces

 In this note we give a criterion for a line bundle on a general blowup of a ruled surface to be k-very ample. (Received 29 November 2000; in revised form 2 April 2001)     

Koszul Duality Patterns in Representation Theory

The aim of this paper is to work out a concrete example as well as to provide the general pattern of applications of Koszul duality to representation theory. The paper consists of three parts relatively independent of each other. The first part gives a reasonably selfcontained introduction to Koszul rings and Koszul duality. Koszul rings are certain -graded rings with particularly nice homological properties which involve a kind of duality. Thus, to a Koszul ring one associates naturally the dual Koszul ring. The second part is devoted to an application to representation theory of semisimple Lie algebras. We show that the block of the Bernstein-Gelfand-Gelfand category that corresponds to any fixed central character is governed by the Koszul ring. Moreover, the dual of that ring governs a certain subcategory of the category again. This generalizes the selfduality theorem conjectured by Beilinson and Ginsburg in 1986 and proved by Soergel in 1990. In the third part we study certain categories of mixed perverse sheaves on a variety stratified by affine linear spaces. We provide a general criterion for such a category to be governed by a Koszul ring. In the flag variety case this reduces to the setup of part two. In the more general case of affine flag manifolds and affine Grassmannians the criterion should yield interesting results about representations of quantum groups and affine Lie algebras.

     

Seshadri constants on rational surfaces with anticanonical pencils

We study a Seshadri constant at a general point on a rational surface whose anticanonical linear system contains a pencil. First, we describe a Seshadri constant of an ample line bundle on such a rational surface explicitly by the numerical data of the ample line bundle. Second, we classify log del Pezzo surfaces which are special in terms of the Seshadri constants of the anticanonical divisors when the anticanonical degree is between 4 and 9.     

Effective estimates on the very ampleness of the canonical line bundle of locally Hermitian symmetric spaces

We study the problem about the very ampleness of the canonical line bundle of compact locally Hermitian symmetric manifolds of non-compact type. In particular, we show that any sufficiently large unramified covering of such manifolds has very ample canonical line bundle, and give estimates on the size of the covering manifold, which is itself a locally Hermitian symmetric manifold, in terms of geometric data such as injectivity radius or degree of coverings.

     

Very Ampleness of Line Bundles and Canonical Embedding of Coverings of Manifolds

Let L be an ample line bundle on a Kähler manifolds of nonpositive sectional curvature with K as the canonical line bundle. We give an estimate of m such that K+mL is very ample in terms of the injectivity radius. This implies that m can be chosen arbitrarily small once we go deep enough into a tower of covering of the manifold. The same argument gives an effective Kodaira Embedding Theorem for compact Kähler manifolds in terms of sectional curvature and the injectivity radius. In case of locally Hermitian symmetric space of noncompact type or if the sectional curvature is strictly negative, we prove that K itself is very ample on a large covering of the manifold.     

Syzygies of projective bundles

In this paper we study defining equations and syzygies among them of projective bundles. We prove that for a given p≥0, if a vector bundle on a smooth complex projective variety is sufficiently ample, then the embedding given by the tautological line bundle satisfies property N_p.     

Tilting generators via ample line bundles

It is known that a tilting generator on an algebraic variety X gives a derived equivalence between X and a certain non-commutative algebra. In this paper, we present a method to construct a tilting generator from an ample line bundle, and construct it in several examples.     

Logarithmic transformations of rigid analytic elliptic surfaces

We give new examples of algebraic elliptic surfaces and non-algebraic rigid analytic elliptic surfaces by means of logarithmic transformations. In the complex analytic case, it is known that all multiple fibers of elliptic surfaces are obtained by logarithmic transformations. Using rigid analytic geometry, we construct similar transformations of elliptic surfaces over complete non-Archimedean valuation base fields. These operations yield rigid analytic elliptic fibrations with multiple fibers. When the resulting surface admits an ample line bundle, we may algebraize the surface. In the positive characteristic case, we obtain new types of algebraic elliptic surfaces. We also obtain a non-algebraic rigid analytic surface the combination of whose invariants appears neither in the algebraic case nor in the complex analytic case.     

On the Seshadri constants of adjoint line bundles

In the present paper we study the possible values of Seshadri constants. While in general every positive rational number appears as the local Seshadri constant of some ample line bundle, we point out that for adjoint line bundles there are explicit lower bounds depending only on the dimension of the underlying variety. In the surface case, where the optimal lower bound is 1/2, we characterize all possible values in the range between 1/2 and 1??there are surprisingly few. As expected, one obtains even more restrictive results for the Seshadri constants of adjoints of very ample line bundles. Finally, we study Seshadri constants of adjoint line bundles in the multi-point setting.     

The absolutely Koszul property of Veronese subrings and Segre products

Absolutely Koszul algebras are a class of rings over which any finite graded module has a rational Poincaré series. We provide a criterion to detect non-absolutely Koszul rings. Combining the criterion with Macaulay2 computations, we identify large families of Veronese subrings and Segre products of polynomial rings which are not absolutely Koszul. In particular, we classify completely the absolutely Koszul algebras among Segre products of polynomial rings, when the base field has characteristic 0.     

Commutative Non-Noetherian Rings with the Diamond Property

A ring R is said to have property (◇) if the injective hull of every simple R-module is locally Artinian. By landmark results of Matlis and Vamos, every commutative Noetherian ring has (◇). We give a systematic study of commutative rings with (◇), We give several general characterizations in terms of co-finite topologies on R and completions of R. We show that they have many properties of Noetherian rings, such as Krull intersection property, and recover several classical results of commutative Noetherian algebra, including some of Matlis and Vamos. Moreover, we show that a complete rings has (◇) if and only if it is Noetherian. We also give a few results relating the (◇) property of a local ring with that of its associated graded rings, and construct a series of examples.

     

Variation of geometric invariant theory quotients

Geometric Invariant Theory gives a method for constructing quotients for group actions on algebraic varieties which in many cases appear as moduli spaces parameterizing isomorphism classes of geometric objects (vector bundles, polarized varieties, etc.). The quotient depends on a choice of an ample linearized line bundle. Two choices are equivalent if they give rise to identical quotients. A priori, there are infinitely many choices since there are infinitely many isomorphism classes of linearized ample line bundles. Hence several natural questions arise. Is the set of equivalence classes, and hence the set of non-isomorphic quotients, finite? How does the quotient vary under change of the equivalence class? In this paper we give partial answers to these questions in the case of actions of reductive algebraic groups on nonsingular projective algebraic varieties. We shall show that among ample line bundles which give projective geometric quotients there are only finitely many equivalence classes. These classes span certain convex subsets (chambers) in a certain convex cone in Euclidean space, and when we cross a wall separating one chamber from another, the corresponding quotient undergoes a birational transformation which is similar to a Mori flip.     

Mumford's example and a general construction

We construct a line bundle on a complex projective manifold (a general ruled variety over a curve) which is not ample, but whose restriction to every proper subvariety is ample. This example is of interest in connection with ampleness questions of vector bundles on varieties of dimension greater than one. The method of construction shows that a stable bundle of positive degree on a curve is ample. The example can be used to show that there is no restriction theorem for Bogomolov stability.     

Koszul incidence algebras, affine semigroups, and Stanley-Reisner ideals

We prove a theorem unifying three results from combinatorial homological and commutative algebra, characterizing the Koszul property for incidence algebras of posets and affine semigroup rings, and characterizing linear resolutions of squarefree monomial ideals. The characterization in the graded setting is via the Cohen-Macaulay property of certain posets or simplicial complexes, and in the more general nongraded setting, via the sequential Cohen-Macaulay property.     

Confluence Algebras and Acyclicity of the Koszul Complex

The N-Koszul algebras are N-homogeneous algebras satisfying a homological property. These algebras are characterised by their Koszul complex: an N-homogeneous algebra is N-Koszul if and only if its Koszul complex is acyclic. Methods based on computational approaches were used to prove N-Koszulness: an algebra admitting a side-confluent presentation is N-Koszul if and only if the extra-condition holds. However, in general, these methods do not provide an explicit contracting homotopy for the Koszul complex. In this article we present a way to construct such a contracting homotopy. The property of side-confluence enables us to define specific representations of confluence algebras. These representations provide a candidate for the contracting homotopy. When the extra-condition holds, it turns out that this candidate works. We make explicit our construction on several examples.     

A Poincaré–Birkhoff–Witt criterion for Koszul operads

The aim of this article is to give a criterion, generalizing the criterion introduced by Priddy for algebras, to prove that an operad is Koszul. We define the notion of Poincaré–Birkhoff–Witt basis in the context of operads. Then we show that an operad having a Poincaré–Birkhoff–Witt basis is Koszul. Besides, we obtain that the Koszul dual operad has also a Poincaré–Birkhoff–Witt basis. We check that the classical examples of Koszul operads (commutative, associative, Lie, Poisson) have a Poincaré–Birkhoff–Witt basis. We also prove by our methods that new operads are Koszul.     

Koszul代数的Hochschild上同调环

基于Buchweitz等人对Koszul代数的Hochschild上同调环的乘法结构的细致分析,给出了Koszul代数的Hochschild上同调环的乘法本质上是平行路的毗连的一个充要条件,并由此重新证明了二次三角string代数的Hochschild上同调环的乘法是平凡的,从而改进了Bustamante的证明.     

The Koszul Complex and Duality

This paper is basically about local, Cohen-Macaulay rings although many of the theorems are more general in nature. The major tool that we shall use is the Koszul complex and its self duality. The Koszul complex is closely linked with the functors Γ and λ which we shall define presently. Our main results will be the computation of the derived functors of Γ and λ.