Linear bounds for levels of stable rationality

Let G be one of the groups SL _n (ℂ), Sp_2n (ℂ), SO _m (ℂ), O _m (ℂ), or G ₂. For a generically free G-representation V, we say that N is a level of stable rationality for V/G if V/G × ℙ ^N is rational. In this paper we improve known bounds for the levels of stable rationality for the quotients V/G. In particular, their growth as functions of the rank of the group is linear for G being one of the classical groups.     

Laminar currents in ℙ<Superscript>2</Superscript>

 In this paper we study laminar currents in ℙ². Given a sequence of irreducible algebraic curves (C _n ) converging in the sense of currents to T, we find geometric conditions on the curves ensuring that the limit current T is laminar. This criterion is then applied to meromorphic dynamical systems in ℙ², and laminarity of the dynamical ``Green' current is obtained for a wide class of meromorphic self maps of ℙ², as well as for all bimeromorphic maps of projective surfaces. Received: 24 September 2001 / Published online: 10 February 2003 Mathematics Subject Classification (2000): 32U40, 37Fxx, 32H50     

On the divisor class group of double solids

For a double solid V→ℙ_3> branched over a surface B⊂ℙ₃(ℂ) with only ordinary nodes as singularities, we give a set of generators of the divisor class group in terms of contact surfaces of B with only superisolated singularities in the nodes of B. As an application we give a condition when H ^* (˜V , ℤ) has no 2-torsion. All possible cases are listed if B is a quartic. Furthermore we give a new lower bound for the dimension of the code of B. Received: 16 November 1998     

On q *-bisimple IC semigroups of type-E

A semigroup is called type-E if the band of its idempotents can be expressed as a direct product of a rectangular band and an ω-chain. For brevity, we call an IC ^*-bisimple quasi-adequate semigroup of type-E a q ^*-bisimple IC semigroup of type-E. In this paper, we characterize q ^*-bisimple semigroups by using some kind of generalized Bruck-Reilly extensions. As a consequence, some results concerning *-bisimple type-A ω-semigroups given by Asibong-Ibe (Semigroup Forum 31:99–117, 1985) are generalized.     

Fibers of the Baum-Bott map for foliations of degree two on ?2

The Baum-Bott map associates to a foliation the Baum-Bott indexes of their singularities. In this paper we study the fibers of the Baum-Bott map in the space of foliations of degree two on the projective plane ℙ². In the main result we prove that its generic fiber contains exactly 240 orbits of the natural action of Aut(ℙ²) onthespace of foliations.     

Infinitesimal extensions of ℙ1 and their Hilbert schemes

 In order to calculate the multiplicity of an isolated rational curve C on a local complete intersection variety X, i.e. the length of the local ring of the Hilbert Scheme of X at [C], it is important to study infinitesimal neighborhoods of the curve in X. This is equivalent to infinitesimal extensions of ℙ¹ by locally free sheaves. In this paper we study infinitesimal extensions of ℙ¹, determine their structure and obtain upper and lower bounds for the length of the local rings of their Hilbert schemes at [ℙ¹]. Received: 11 June 2001 / Revised version: 28 January 2002     

Generalizations of the odd degree theorem and applications

LetV ⊂ ℙℝ ⁿ be an algebraic variety, such that its complexificationV _ℂ ⊂ ℙ ⁿ is irreducible of codimensionm ≥ 1. We use a sufficient condition on a linear spaceL ⊂ ℙℝ ⁿ of dimensionm + 2r to have a nonempty intersection withV, to show that any six dimensional subspace of 5 × 5 real symmetric matrices contains a nonzero matrix of rank at most 3.     

Resolutions of generic points lying on a smooth quadric

For a 0-dimensional schemeX on a smooth quadricQ we define a special type of resolution of its ideal sheaf as a locally freeO _Q. These resolutions allow to find, for schemes which are generic inQ, the minimal free resolution ofX as a subscheme of ℙ³. For almost all such schemes the graded Betti numbers in ℙ³ depend only on the Hilbert function ofX in ℙ³. Work done with financial support of M.U.R.S.T., while the authors were members of C.N.R.     

Index parity of closed geodesics and rigidity of Hopf fibrations

The diameter rigidity theorem of Gromoll and Grove [1987] states that a Riemannian manifold with sectional curvature ≥ 1 and diameter ≥ π/2 is either homeomorphic to a sphere, locally isometric to a rank one symmetric space, or it has the cohomology ring of the Cayley plane Caℙ. The reason that they were only able to recognize the cohomology ring of Caℙ is due to an exceptional case in another theorem [Gromoll and Grove, 1988]: A Riemannian submersion σ:?^m→B ^b with connected fibers that is defined on the Euclidean sphere ?^m is metrically congruent to a Hopf fibration unless possibly (m,b)=(15,8). We will rule out the exceptional cases in both theorems. Our argument relies on a rather unusal application of Morse theory. For that purpose we give a general criterion which allows to decide whether the Morse index of a closed geodesic is even or odd. Oblatum 7-II-2000 & 11-X-2000?Published online: 29 January 2001     

Weighted integral formulas on manifolds

We present a method of finding weighted Koppelman formulas for (p,q)-forms on n-dimensional complex manifolds X which admit a vector bundle of rank n over X×X, such that the diagonal of X×X has a defining section. We apply the method to ℙ ⁿ and find weighted Koppelman formulas for (p,q)-forms with values in a line bundle over ℙ ⁿ . As an application, we look at the cohomology groups of (p,q)-forms over ℙ ⁿ with values in various line bundles, and find explicit solutions to the -equation in some of the trivial groups. We also look at cohomology groups of (0,q)-forms over ℙ ⁿ ×ℙ ^m with values in various line bundles. Finally, we apply our method to developing weighted Koppelman formulas on Stein manifolds.     

Steinberg triality groups acting on 2 ? (<Emphasis Type="Italic">v,k</Emphasis>, 1) designs

A 2 - (υ, k, 1) design D = (ℙ,ℙ, ℬ) is a system consisting of a finite set ℙ of υ points and a collection ℬ of ℙ-subsets of ℙ, called blocks, such that each 2-subset of ℙ is contained in precisely one block. Let G be an automorphism group of a 2-(υ, k, 1) design. Delandtsheer proved that if G is block-primitive and D is not a projective plane, then G is almost simple, that is, T ⩽ G ⩽ Aut(T), where T is a non-abelian simple group. In this paper, we prove that T is not isomorphic to ³ D ₄(q). This paper is part of a project to classify groups and designs where the group acts primitively on the blocks of the design.     

Asymptotic purity for very general hypersurfaces of ℙ<Superscript><Emphasis Type="Italic">n</Emphasis></Superscript> × ℙ<Superscript><Emphasis Type="Italic">n</Emphasis></Superscript> of bidegree (<Emphasis Type="Italic">k,k</Emphasis>)

For a complex irreducible projective variety, the volume function and the higher asymptotic cohomological functions have proven to be useful in understanding the positivity of divisors as well as other geometric properties of the variety. In this paper, we study the vanishing properties of these functions on hypersurfaces of ℙ ⁿ × ℙ ⁿ . In particular, we show that very general hypersurfaces of bidegree (k, k) obey a very strong vanishing property, which we define as asymptotic purity: at most one asymptotic cohomological function is nonzero for each divisor. This provides evidence for the truth of a conjecture of Bogomolov and also suggests some general conditions for asymptotic purity.     

Mumford dendrograms and discrete p-adic symmetries

In this article, we present an effective encoding of dendrograms by embedding them into the Bruhat-Tits trees associated to p-adic number fields. As an application, we show how strings over a finite alphabet can be encoded in cyclotomic extensions of ℚ _p and discuss p-adic DNA encoding. The application leads to fast p-adic agglomerative hierarchic algorithms similar to the ones recently used e.g. by A. Khrennikov and others. From the viewpoint of p-adic geometry, to encode a dendrogram X in a p-adic field K means to fix a set S of K-rational punctures on the p-adic projective line ℙ₁. To ℙ₁ \ S is associated in a natural way a subtree inside the Bruhat-Tits tree which recovers X, a method first used by F. Kato in 1999 in the classification of discrete subgroups of PGL₂(K). Next, we show how the p-adic moduli space of ℙ¹ with n punctures can be applied to the study of time series of dendrograms and those symmetries arising from hyperbolic actions on ℙ¹. In this way, we can associate to certain classes of dynamical systems a Mumford curve, i.e. a p-adic algebraic curve with totally degenerate reduction modulo p. Finally, we indicate some of our results in the study of general discrete actions on ℙ¹, and their relation to p-adic Hurwitz spaces. The text was submitted by the author in English.     

Decomposability criterion for linear sheaves

We establish a decomposability criterion for linear sheaves on ℙ ⁿ . Applying it to instanton bundles, we show, in particular, that every rank 2n instanton bundle of charge 1 on ℙ ⁿ is decomposable. Moreover, we provide an example of an indecomposable instanton bundle of rank 2n − 1 and charge 1, thus showing that our criterion is sharp.     

Subdirectly irreducible \mathcal{RGC} -commutative right H-semigroups

A semigroup S is said to be ℛ-commutative if, for all elements a,b∈S, there is an element x∈S ¹ such that ab=bax. A semigroup S is called a generalized conditionally commutative (briefly, -commutative) semigroup if it satisfies the identity aba ²=a ² ba. An ℛ-commutative and -commutative semigroup is called an -commutative semigroup. A semigroup S is said to be a right H-semigroup if every right congruence of S is a congruence of S. In this paper we characterize the subdirectly irreducible semigroups in the class of -commutative right H-semigroups. Research supported by the Hungarian NFSR grant No T029525.     

Symplectic 4-manifolds as branched coverings of ℂℙ2

We show that every compact symplectic 4-manifold X can be topologically realized as a covering of ℂℙ² branched along a smooth symplectic curve in X which projects as an immersed curve with cusps in ℂℙ². Furthermore, the covering map can be chosen to be approximately pseudo-holomorphic with respect to any given almost-complex structure on X. Oblatum 9-III-1999 & 2-IX-1999 / Published online: 29 November 1999     

Double rational normal curves with linear syzygies

In this note we are looking after nilpotent projective curves without embedded points, which have rational normal curves of degree d as support, are defined (scheme-theoretically) by quadratic equations, have degree 2d and have only linear syzygies. We show that, as expected, no such curve does exist in ℙ ^d , and then consider doublings in a bigger ambient space. The simplest and trivial example is that of a double line in the plane. We show that the only possibility is to take rational normal curves in ℙ ^d embedded further in ℙ^{2 d} and to take a certain doubling in the sense of Ferrand (cf. [5]) in ℙ^{2 d} . These double curves have the Hilbert polynomial H(t)=2dt+1, i.e. they are in the Hilbert scheme of the rational normal curves of degree 2d. Thus, it turns out that they are natural generalizations of the double line in the plane considered as a degenerated conic.The simplest nontrivial example is the curve of degree 4 in ℙ⁴, defined by the ideal (xz−y ², xu−yv, yu−zv, u ², uv, v ²). The double rational curve allow the formulation of a Strong Castelnuovo Lemma in the sense of [7], for sets of points and double points. In the last section we mention some plethysm formulae for symmetric powers. Received: 1 September 2000 / Revised version: 15 January 2001     

The diffeomorphic types of the complements of arrangements in ??3 II

For any arrangement of hyperplanes in ℂℙ³, we introduce the soul of this arrangement. The soul, which is a pseudo-complex, is determined by the combinatorics of the arrangement of hyperplanes. In this paper, we give a sufficient combinatoric condition for two arrangements of hyperplanes to be diffeomorphic to each other. In particular we have found sufficient conditions on combinatorics for the arrangement of hyperplanes whose moduli space is connected. This generalizes our previous result on hyperplane point arrangements in ℂℙ³. This work was partially supported by NSA grant and NSF grant     

On monomial k-Buchsbaum curves in ℙ3

Using the theory of affine semigroup rings we describe all strictly 2-Buchsbaum monomial curves in ℙ³.     

On Buchsbaum bundles on quadric hypersurfaces

Let E be an indecomposable rank two vector bundle on the projective space ℙ ⁿ , n ≥ 3, over an algebraically closed field of characteristic zero. It is well known that E is arithmetically Buchsbaum if and only if n = 3 and E is a null-correlation bundle. In the present paper we establish an analogous result for rank two indecomposable arithmetically Buchsbaum vector bundles on the smooth quadric hypersurface Q _n ⊂ ℙ ⁿ⁺¹, n ≥ 3. We give in fact a full classification and prove that n must be at most 5. As to k-Buchsbaum rank two vector bundles on Q ₃, k ≥ 2, we prove two boundedness results.