On 2-spannedness for the adjunction mapping

LetL be a line bundle on a smooth connected projective manifold X of dimension n. We extend to any dimension the definition of k-spannedness forL; this is a notion of k-th order embedding which was recently given in the case of curves and surfaces. Then, by a reduction to the surfaces case, we prove that the adjoint bundle K_x+(n–1)L is 2-spanned ifL is (at least) 3-spanned.     

Effective adjunction theory

Here we investigate the property of effectivity for adjoint divisors. Among others, we prove the following results: A projective variety X with at most canonical singularities is uniruled if and only if for each very ample Cartier divisor H on X we have \(H^0(X, m_0K_X+H)=0\) for some \(m_0=m_0(H)>0\). Let X be a projective 4-fold, L an ample divisor and t an integer with \(t \ge 3\). If \(K_X+tL\) is pseudo-effective, then \(H^0(X, K_X+tL) \ne 0\).     

Infinitesimal adjunction and polar curves

The polar curves of foliations having a curve C of separatrices generalize the classical polar curves associated to hamiltonian foliations of C. As in the classical theory, the equisingularity type ℘() of a generic polar curve depends on the analytical type of , and hence of C. In this paper we find the equisingularity types ε(C) of C, that we call kind singularities, such that ℘() is completely determined by ε(C) for Zariski-general foliations . Our proofs are mainly based on the adjunction properties of the polar curves. The foliation-like framework is necessary, otherwise we do not get the right concept of general foliation in Zariski sense and, as we show by examples, the hamiltonian case can be out of the set of general foliations. The author was partially supported by the research projects MTM2007-66262 (Ministerio de Educación y Ciencia), MTM2006-15338-C02-02 (Ministerio de Educación y Ciencia),VA059A07 (Junta de Castilla y León) and PGIDITI06PXIB377128PR (Xunta de Galicia).     

Injective spaces via adjunction

The work of the present author and his coauthors over the past years gives evidence that it may be useful to regard each topological space as a kind of enriched category, by interpreting the convergence relation x→x between ultrafilters and points of a topological space X as arrows in X. Naturally, this point of view opens the door to the use of concepts and ideas from enriched Category Theory for the investigation of topological spaces. Topological theories introduced by the author provide a convenient general setting for appropriately transferring these concepts and ideas to the world of topological spaces and some other geometric objects such as approach spaces. Using tools like adjunction and the Yoneda lemma, we show that the cocomplete spaces are precisely the injective spaces, and they are algebras for a suitable monad on . This way we obtain enriched versions of known results about injective topological spaces and continuous lattices.     

On the adjunction process over a surface in char.p

Let K_s be the canonical bundle on a non singular projective surface S (over an algebraically closed field F, char F=p) and L be a very ample line bundle on S. Suppose (S,L) is not one of the following pairs: (P²,O(e)), e=1,2, a quadric, a scroll, a Del Pezzo surface, a conic bundle. Then

1. (K_s?L)² is spanned at each point by global sections. Let \(\phi :S \to P^N _F \) be the map given by the sections Γ(K_s?L)², and let φ=s o r its Stein factorization.
2. r:S→S′=r(S) is the contraction of a finite number of lines, E_i for i=1,...r, such that E_i·E_i=K_S·E_i=?L·E_i=?1.
3. If h°(L)≥6 and L·L≥9, then s is an embedding.

     

On regular le-semigroups

Communicated by J. S. Ponizovski?     

On normal and regular identities

We deal with two kinds of special identities: normal and regular, considered by Mel'nik, Ponka and other authors. We point out fundamental properties of these identities. Also in §2 we show that the lattice of all subvarieties of the variety defined by all normal identities of a given varietyV (called the normal part of a varietyV) is isomorphic to the direct product of the lattice L(V) and a two-element chain. This result (Theorem 3) is a strengthening of a result of Mel'nik [14]. Theorem 4 states that the word problem for free algebras of the variety defined by all normal identities ofV is solvable if and only if it is solvable forV, which is due to the property of regular identities, proved in [8]. In §3 we consider normal and regular consequences of a given set of identities. Theorem 6 shows that for a given varietyV, satisfying a nonregular absorption law, the lattice L(Mod(NR(V))) is isomorphic to the direct product of the lattice L(V) and a four-element lattice, with two atoms.Theorems in §4 collect some of results on the existence of a finite basis for normal and regular part of a given, finitely presented varietyV and of the finite basis property, as well, strengthening the result of Lakser, Padmanabhan and Platt [12].Results above can be applied for semigroup varieties.Presented by George Grätzer.     

On regular rings and annihilators

Let F_n stand for the distribution of a normalized sum of n independent random variables with common distribution H. In [6] we assumed the restricted convergence. and obtain an analogous result. The method of proof is considerably different, in particular a very recent continuation theorem (lemma 3.2) for infinitely divisible distributions is needed.     

On regular rings and V-rings

In this note, certain generalisations of strongly regular rings are considered in connection with regular rings andV-rings. The result that strongly regular rings are left (and right)V-rings [11] is extended. A condition for prime leftV-rings to be primitive with non-zero socle is given (this is related to a question ofFisher [7, Problem 3]. IfA is an ALD (almost left duo) ring, then (1) a simple leftA-module is injective iff it isp-injective; (2)A is von Neumann regular iff every maximal essential right ideal ofA isf-injective. Characterisations of semi-simple Artinian and simple Artinian rings are given in terms of regular andV-rings.