Volume 5, Issue 2

Let D ( n, s, k ) = { z ∈ ℂ n : ( –| z 1 | 2  + | z 2 | 2 ) k + ∑ i s =3 | z i | 2 – ∑ i n = s +1 | z i | 2 < 1}, where k  ∈ ℤ + , 3 ≤ s ≤ n , and let M ( n , s , k ) = ∂ D ( n , s , k ). We determine for any values of n , s and k the biholomorphic automorphisms of D ( n , s , k ), identifying, in suitable cases, the germs of biholomorphisms that map M ( n , s , k ) and D ( n , s , k ) respectively in themselves. It turns out that, if k is even and s  = n , any such germ extends to a biholomorphic automorphism of D ( n , s , k ), while, in other cases, it is a branch of an algebraic correspondence.

Virtual algebraic Betti numbers for real algebraic varieties are constructed from the property of additivity of d -algebraically constructible homologies.

We solve the long-standing open problem of classifying all 3-( v , k , 1) designs with a flag-transitive group of automorphisms (cf. [11], p. 147, and [12], p. 273, but presumably dating back to 1965). Our result relies on the classification of the finite 2-transitive permutation groups.

We give sharp lower bounds for the postulation of the nodes of a general plane projection of a smooth connected curve C ⊆ ℙ r and we study the relationships with the geometry of the embedding. Strict connections with Castelnuovo’s theory and Halphen’s theory are shown.

We consider the question whether the system of lines of a two-dimensional topological plane can be described as the system of geodesics of a Riemannian metric or an affine connection. In [4] we have shown that non-classical compact projective planes do not admit Riemannian metrics. Here we use different methods that are suited to two-dimensional planes in general. We apply them to three families of topological affine planes with large collineation groups. It turns out that for the standard models of the skew-parabola planes and the planes over cartesian fields no affine connection, and hence no Riemannian metric exist. However, for the Moulton planes affine connections do exist and we determine all of them. Among them is at least one Riemannian connection. We also give an example where no Riemannian metric exists. Moreover, we derive a characterization of the classical affine plane in the case that ℝ 2 acts by vector space translations as a subgroup of the collineation group.

A (0, α )-geometry, α ≥ 3, fully embedded in AG( n , q ) is always a linear representation[2]. In [4] the (0, 2)-geometries fully embedded in AG(3, q ) are classified up to linear representations. Here we extend this result to full embeddings in AG( n , q ). As a corollary we classify the semipartial geometries with α > 1 fully embedded in AG( n , q ), but not as a linear representation.

In this paper, we address three problems in the theory of incidence structures associated with isoparametric hypersurfaces with four distinct principal curvatures in spheres. In [4], we showed that these isoparametric hypersurfaces and their focal manifolds yield Tits buildings of type C 2 . Here, we first give a short proof for the description of point rows and line pencils in these incidence structures. Furthermore, we provide criteria which characterize intersecting lines (or points which can be joined by a line, respectively) and obtain a simple description of the intersection point (the joining line, respectively). Finally, we describe for each antiflag ( p , L ) the position of the uniquely determined flag ( q , K  ) such that ( p ,  K ,  q ,  L ) is a 3-chain. In this paper, we make extensive use of the theory of isoparametric triple systems as developed in [1].

The paper presents a series of principally different C ∞ -smooth counterexamples to the following hypothesis on a characterization of the sphere: Let K ⊂ ℝ 3 be a smooth convex body. If at every point of ∂ K , we have R 1 ≤ C ≤ R 2 for a constant C , then K is a ball. ( R 1 and R 2 stand for the principal curvature radii of ∂ K .) The hypothesis was proved by A. D. Alexandrov and H. F. Münzner for analytic bodies. For the case of general smoothness it has been an open problem for years. Recently, Y. Martinez-Maure has presented a C 2 -smooth counterexample to the hypothesis.

On a complex surface with a cone singularity, solutions to parabolic equations may achieve non-smooth extrema. Using a barrier function allows application of the maximum principle, yielding the same estimates as in the smooth case.

Let C be a convex body. By C -length of a segment we mean the ratio of the length of this segment to the half of the length of a longest parallel chord of C . We show that every planar convex body C permits an inscribed triangle whose sides are of equal C -length at least 3/2. Equivalently, C can be packed with three equal homothetical copies of ratio at least 3/7 which touch the boundary of C and touch pairwise themselves.