Band 10, Heft 3

In this paper we show the following property of a non Levi flat real hypersurface in : if the unit characteristic direction T is a geodesic, then it is an eigenvector of the second fundamental form and the relative eigenvalue is constant. As an application we prove a symmetry result of Alexandrov type for compact hypersurfaces in with positive constant Levi mean curvature.

We prove lower bounds on the size of small maximal partial spreads in Q + (4 n + 1, q ), W (2 n + 1, q ), and H (2 n + 1, q 2 ). This research on the size of smallest maximal partial spreads in classical finite polar spaces is part of a detailed study on small and large maximal partial ovoids and spreads in classical finite polar spaces, performed in [De Beule, Klein, Metsch, Storme, Des. Codes Cryptogr 47: 21–34, 2008, De Beule, Klein, Metsch, Storme, European J. Combin 29: 1280–1297, 2008].

We give several results concerning the notion of antipodes (i.e., farthest points) on the surface of a polyhedron endowed with its intrinsic metric.

A well-known conjecture asserts that smooth threefolds are quadratically normal with the only exception of the Palatini scroll. As a corollary of a more general statement we obtain the following result, which is related to the previous conjecture: If is not quadratically normal, then its triple curve is reducible. Similar results are also given for higher dimensional varieties.

We introduce a family of algebraic curves over 𝔽 q 2 n (for an odd n ) and show that they are maximal. When n = 3, our curve coincides with the 𝔽 q 6 -maximal curve that has been found by Giulietti and Korchmáros recently. Their curve (i.e., the case n = 3) is the first example of a maximal curve proven not to be covered by the Hermitian curve.

The classical Julia–Wolff–Carathéodory Theorem is one of the main tools to study the boundary behavior of holomorphic self-maps of the unit disc of ℂ. In this paper we prove a Julia–Wolff–Carathéodory type theorem in the case of the polydisc of . The Busemann functions are used to define a class of “generalized horospheres” for the polydisc and to extend the notion of non-tangential limit. With these new tools we give a generalization of the classical Julia Lemma and of the Lindelöf Theorem, which the new Julia–Wolff–Carathéodory Theorem relies upon.

In this paper, we show that the building at infinity of a two-dimensional affine ℝ-building is a generalized polygon endowed with a valuation satisfying some specific axioms. Specializing to the discrete case of affine buildings, this solves part of a long standing conjecture about affine buildings of type , and it reproves the results obtained mainly by the second author for types and . The techniques are completely different from the ones employed in the discrete case, but they are considerably shorter, and general (i.e., independent of the type of the two-dimensional ℝ-building).

It is a classical question in convex geometry whether there exists a σ -finite Borel measure on the space of convex bodies such that every open subset has positive measure. In this paper we give a positive answer by introducing a class of measures with the desired feature and additional invariance properties which are (in a certain sense) strongest possible.

We consider compact hypersurfaces in an ( n + 1)-dimensional either Riemannian or Lorentzian space endowed with a conformal Killing vector field. For such hypersurfaces, we establish an integral formula which, especially in the simpler case when is a product space, allows us to derive some interesting consequences in terms of the scalar curvature of the hypersurface. For instance, when n = 2 and is either the sphere or the real projective plane , we characterize the slices of the trivial totally geodesic foliation as the only compact two-sided surfaces with constant Gaussian curvature in the Riemannian product such that its angle function does not change sign. When n ≥ 3 and is a compact Einstein Riemannian manifold with positive scalar curvature, we also characterize the slices as the only compact two-sided hypersurfaces with constant scalar curvature in the Riemannian product whose angle function does not change sign. Similar results are also established for spacelike hypersurfaces in a Lorentzian product 𝕄 × ℝ 1 .

Based on the first author's diploma thesis [Kurth, On a real form of E 6 and its related generalized quadrangle, 2000] we use the theories of Lie groups and of Tits buildings in order to describe a Veronese embedding of the real quadrangle of type E 6 , i.e., the C 2 sub-building of the complex E 6 building corresponding to the real form E 6(–14) of the semisimple complex Lie group of type E 6 .

In this paper we study the linear series |L – 3 p | of hyperplane sections with a triple point p on a surface S embedded via a very ample line bundle L for a general point p . If this linear series does not have the expected dimension we call ( S, L ) triple-point defective . We show that on a triple-point defective surface through a general point every hyperplane section has either a triple component or the surface is rationally ruled and the hyperplane section contains twice a fibre of the ruling.

We study minimal complex surfaces S of general type with q ( S ) = q and p g ( S ) = 2 q – 3, q ≥ 5. We give a complete classification in case that S has a fibration onto a curve of genus ≥ 2. For these surfaces K 2 = 8 χ . In general we prove that K 2 ≥ 7 χ – 1 and that the stronger inequality K 2 ≥ 8 χ holds under extra assumptions (e.g., if the canonical system has no fixed part or the canonical map has even degree). We also describe the Albanese map of S .

An ideal membership test is presented by A. Leykin in Numerical primary decomposition using dual bases. A counter example is presented to demonstrate that the proposed test is invalid.