On the sectional geometric genus of quasi-polarized varieties,II

Let (X,L) be a quasi-polarized variety of dim X=n. In the previous paper we gave a new invariant (the i-th sectional geometric genus) of (X,L), which is a generalization of the degree and the sectional genus of (X,L). In this paper we study some properties of the sectional geometric genus, and we consider the i-th sectional geometric genus of some special varieties. This research was partially supported by Grant-in-Aid for Scientific Research (No.14740018) from the Ministry of Education, Culture, Sports, Science and Technology.Mathematics Subject Classification (2000):Primary 14C20; Secondary 14J30, 14J32, 14J35, 14J40, 14J45, 14N99     

A simply connected surface of general type with <Emphasis Type="Italic">p</Emphasis><Subscript><Emphasis Type="Italic">g</Emphasis></Subscript>=0 and <Emphasis Type="Italic">K</Emphasis><Superscript>2</Superscript>=2

In this paper we construct a simply connected, minimal, complex surface of general type with p _g =0 and K ²=2 using a rational blow-down surgery and a ℚ-Gorenstein smoothing theory. Mathematics Subject Classification (2000) Primary 14J29; Secondary 14J10, 14J17, 53D05     

Short proofs for two theorems of Chien,Hell and Zhu

In (J Graph Theory 33 (2000), 14–24), Hell and Zhu proved that if a series–parallel graph G has girth at least 2?(3k?1)/2?, then χ_c(G)≤4k/(2k?1). In (J Graph Theory 33 (2000), 185–198), Chien and Zhu proved that the girth condition given in (J Graph Theory 33 (2000), 14–24) is sharp. Short proofs of both results are given in this note. © 2010 Wiley Periodicals, Inc. J Graph 66: 83‐88, 2010 Theory     

On the second sectional H-arithmetic genus of polarized manifolds

Let (X,L) be a polarized variety defined over the complex number field with dim X=n. In this paper we introduce the notion of the i-th sectional H-arithmetic genus ^H_i(X,L) for every integer i with 0in. We expect that this invariant has a property similar to the Euler-Poincaré characteristic of the structure sheaf of i-dimensional varieties. In this paper, we consider the case where X is smooth and i=2, and we study a polarized version of some results in the theory of surfaces.Mathematics Subject Classification (2000): 14C20, 14C17, 14C40, 14J29, 14J30, 14J35, 14J40This research was partially supported by the Grant-in-Aid for Young Scientists (B) (No.14740018), The Ministry of Education, Culture, Sports, Science and Technology, Japan.     

Characterization theorems for the projective space and vector bundle adjunction

 We consider some conditions under which a smooth projective variety X is actually the projective space. We also extend to the case of positive characteristic some results in the theory of vector bundle adjunction. We use methods and techniques of the so called Mori theory, in particular the study of rational curves on projective manifolds. Received: 16 May 2002 / Revised version: 18 November 2002 Published online: 3 March 2003 Mathematics Subject Classification (2000): 14E30, 14J40, 14J45     

Nef dimension of minimal models

We reduce the Abundance Conjecture in dimension 4 to the following numerical statement: if the canonical divisor K is nef and has maximal nef dimension, then K is big. From this point of view, we classify in dimension 2 nef divisors which have maximal nef dimension, but which are not big.Mathematics Subject Classification (2000): 14E30, 14J10, 14N30Revised version: 2 March 2004     

On quartics with three-divisible sets of cusps

 We study the geometry and codes of quartic surfaces with many cusps. We apply Gr?bner bases to find examples of various configurations of cusps on quartics. Received: 10 May 2002 / Revised version: 11 November 2002 Published online: 3 March 2003 Permanent address: Institute of Mathematics, Jagiellonian University, ul. Reymonta 4, 30-059 Kraków, Poland. e-mail: rams@mi.uni-erlangen.de, rams@im.uj.edu.pl Research partially supported by the Schwerpunktprogramm ``Global methods in complex geometry' of the Deutsche Forschungsgemeinschaft, and by EAGER. Mathematics Subject Classification (2000): 14J25, 14J17     

Maximally irregularly fibred surfaces of general type

We generalise a method of Xiao Gang to construct prototypes of fibred surfaces with maximal irregularity without being a product. This enables us, in the case of fibre genus g=3 to describe the possible singular fibres and to calculate the invariants of these surfaces. We also prove structure theorems on the moduli space for fibred surfaces with fibre genus g=2 and g=3.Mathematics Subject Classification (2000): 14J10, 14J29, 14D06Acknowledgement The author thanks his thesis advisor F. Herrlich for many stimulating discussions and a lot of patience. He also thanks E. Viehweg for worthful remarks concerning Torellis theorem. Some results in the same direction were obtained independently by J.-X. Cai. The author thanks him and the referee for his suggestions.     

An addendum on algebraic models of smooth manifolds

Here we prove that every compact C manifold has uncountably many algebraic models (a result conjectured in J. Bochnak and W. Kucharz Algebraic models of smooth manifolds, Invent. Math. 97 (1989), 585–611, and independently proved in a different way by J. Bochnak).     

The spectrum for large sets of idempotent quasigroups

In this article we construct a large set of idempotent quasigroups of order 14. Combined with the results in Chang, JCMCC; and Teirlinck and Lindner, Eur J Combin 9 (1988), 83–89, this shows that the spectrum for large sets of idempotent quasigroups of order n [briefly, LIQ(n)] is the set all integers n≥ 3 with the exception of n=6. © 2000 John Wiley & Sons, Inc. J Combin Designs 8: 79–82, 2000     

A characterization of certain irreducible symplectic 4-folds

 We give a characterization of irreducible symplectic fourfolds which are given as Hilbert scheme of points on a K3 surface. Received: 26 June 2002 / Revised version: 9 September 2002 Published online: 14 February 2003 Mathematics Subject Classification (2000): Primary 14J32, Secondary 32Q20     

Non-special scrolls with general moduli

Abstract  In this paper we study smooth, non-special scrolls S of degree d, genus g ≥ 0, with general moduli. In particular, we study the scheme of unisecant curves of a given degree on S. Our approach is mostly based on degeneration techniques. Keywords Ruled surfaces, Hilbert schemes of scrolls, Moduli, Embedded degenerations Mathematics Subject Classification (2000) 14J26, 14D06, 14C20, (Secondary) 14H60, 14N10     

Singular abelian covers of algebraic surfaces

In this paper we generalise F. Cataneses work on singular (/2)²-covers to arbitrary finite abelian covers of algebraic surfaces. We determine the contribution of singularities to the invariants , K ², q, p _g , P _n and the canonical sheaf. We use these computations to construct a surface of general type with birational canonical map ₁, p _g =4 and K ² =31. Mathematics Subject Classification (2000):14J29, 14J17, 14J25, 14E20     

The resolution of the anti‐mitre Steiner triple system conjecture

This paper gives a proof of the existence of anti‐mitre Steiner triple systems of order n for every n ≡ 1,3 mod 6 except for n = 9. © 2005 Wiley Periodicals, Inc. J Combin Designs 14: 229–236, 2006     

Manifolds polarized by vector bundles

LetX be a complex projective manifold of dimension n and let ε be an ample vector bundle of rank r. Let also τ = τ (X,ε) = min {t ∈ ℝ : K_X + t det ε is nef} be the nef value of the pair (X, ε). In this paper we classify the pairs (X, ε) such that{ Mathematics Subject Classification (2000)14J60; 14J40; 14E30     

An Analog of the Milnor Group of a Link in the Higher-Dimensional Link Theory

Singular links consisting of several k-spheres and a p-sphere in the (2k+1)-sphere are studied. For k>1, a theory of such singular links is constructed, which is parallel to J. Milnor's theory of one-dimensional links. Bibliography: 14 titles.     

Existence and multiplicity of nontrivial solutions for an elliptic equation on ℝ<Superscript><Emphasis Type="Italic">N</Emphasis></Superscript> with indefinite linear part

 Via the Linking Theorem and Pseudo-index theory, we consider the existence and multiplicity of nontrivial solutions for a class of elliptic problems in all of ℝ ^N with indefinite linear part involving resonance and non-resonance at any eigenvalue. Received: 9 September 2002 / Revised version: 14 February 2003 Published online: 24 April 2003 Mathematics Subject Classification (2000): 35J20, 35J70     

On isomorphic factorizations of circulant graphs

We investigate the conjecture that every circulant graph X admits a k‐isofactorization for every k dividing |E(X)|. We obtain partial results with an emphasis on small values of k. © 2006 Wiley Periodicals, Inc. J Combin Designs 14: 406–414, 2006     

Five-torsion in the homology of the matching complex on 14 vertices

J.L. Andersen proved that there is 5-torsion in the bottom nonvanishing homology group of the simplicial complex of graphs of degree at most two on seven vertices. We use this result to demonstrate that there is 5-torsion also in the bottom nonvanishing homology group of the matching complex on 14 vertices. Combining our observation with results due to Bouc and to Shareshian and Wachs, we conclude that the case n=14 is exceptional; for all other n, the torsion subgroup of the bottom nonvanishing homology group has exponent three or is zero. The possibility remains that there is other torsion than 3-torsion in higher-degree homology groups of when n≥13 and n≠14. Research of J. Jonsson was supported by European Graduate Program “Combinatorics, Geometry, and Computation”, DFG-GRK 588/2.