Volume 8, Issue 2

We solve a more general version of a conjecture of D. Hachenberger [D. Hachenberger, Groups admitting a Kantor family and a factorized normal subgroup. Des. Codes Cryptogr. 8 (1996), 135–143. MR1393979 (98c:20042) Zbl 0877.51007] by showing that a group admitting a Kantor family with a thick F -factor is a p -group for some prime p .

A Helly-type theorem previously established for a finite family of simply connected orthogonal polygons may be extended to a family of planar compact sets having connected complements: If every three (not necessarily distinct) members of have a nonempty intersection that is starshaped via orthogonally convex paths, then all members of have such an intersection. When is finite, an analogous result holds with orthogonally convex paths replaced by staircase paths. The number three is best possible in each case. Moreover, the results fail without the requirement that the sets have connected complements.

In this paper we improve a classical result of Van de Ven which characterizes linear subspaces of as the only smooth closed subvarieties of for which the normal sequence splits (see [A. Van de Ven, A property of algebraic varieties in complex projective spaces. In: Colloque Géom. Diff. Globale ( Bruxelles , 1958), 151–152, Centre Belge Rech. Math., Louvain 1959. MR0116361 (22 #7149) Zbl 0092.14004]). Precisely we prove the following: Let X be a submanifold of of dimension ≥ 3, and Y ⊆ X a submanifold of X of dimension ≥ 2. Assume that Span( Y ) = Span( X ), where Span( X ) is the smallest linear subspace of containing X . Then the exact sequence: splits if and only if X is a linear subspace of .

Let h 1 ,…, h s ∈ ℝ[ X, Y ] and assume that the set W ( h ) := {( a, b ) ∈ ℝ 2 | h i ( a, b ) ≥ 0 for all 1 ≤ i ≤ s } is compact and non-empty. We give an effective method to decide from the knowledge of h 1 ,…, h s whether every polynomial f ∈ ℝ[ X, Y ], strictly positive on W ( h ), has a representation f = σ 0 + h 1 σ 1 +···+ h s σ s with each σ i being a sum of squares in ℝ[ X, Y ].

We define a topological invariant of complex projective plane curves. As an application, we present new examples of arithmetic Zariski pairs.

Let X be a smooth complex projective variety endowed with an ample vector bundle ε admitting a global section whose zero locus is a smooth subvariety Z of the expected dimension, and let H be an ample line bundle on X , whose restriction H Z to Z is very ample. Triplets ( X, ε, H ) are studied and classified under the assumption that the delta genus of ( Z, H Z ) is either small (≤ 3) or small in comparison with the corank of ε or the degree.

We define defect for hypersurfaces with A-D-E singularities in complex projective normal Cohen–Macaulay fourfolds having some vanishing properties of Bott-type and prove formulae for Hodge numbers of big resolutions of such hypersurfaces. We compute Hodge numbers of Calabi–Yau manifolds obtained as small resolutions of cuspidal triple sextics and double octics with higher A j singularities.

We study dominant rational maps from a general surface in to surfaces of general type. We prove restrictions on the target surfaces, and special properties of these rational maps. We show that for small degree the general surface has no such map. Moreover a slight improvement of a result of Catanese, on the number of moduli of a surface of general type, is also obtained.

The radial centre of a convex body A in ℝ n is the point in A at which the mean value of distances from the boundary of A , with respect to all directions, attains its maximum. We prove that if A is smooth, then its radial centre is an interior point of A .