Volume 11, Issue 3

We correct the definition and the list of all reducible Veronese surfaces in our previous paper “Reducible Veronese surfaces”, Adv. Geom. 10 (2010), 719–735.

A John function is a continuously differentiable function whose gradient is bounded by the reciprocal of the Euclidean distance to the boundary of the domain. Here we construct John functions whose gradient norms have positive lower bounds and whose domain is a definable set of an o-minimal structure. We prove their definability and give explicit estimates for a number of John functions required for a given set to obtain uniform positive lower bounds.

In [Ball, Ebert, Lavrauw, J. Algebra 311: 117–129, 2007] a geometric construction was given of a finite semifield from a certain configuration of two subspaces with respect to a Desarguesian spread in a finite-dimensional vector space over a finite field. Moreover, it was proved that any finite semifield can be obtained in this way. In [Lavrauw, Finite Fields Appl. 14: 897–910, 2008] we proved that the configuration needed for the geometric construction given in [Ball, Ebert, Lavrauw, J. Algebra 311: 117–129, 2007] for finite semifields is equivalent with an ( n – 1)-dimensional subspace skew to a determinantal hypersurface in PG( n 2 – 1, q ), and provided an answer to the isotopism problem in [Ball, Ebert, Lavrauw, J. Algebra 311: 117–129, 2007]. In this paper we give a generalisation of the BEL-construction using linear sets, and then concentrate on this configuration and the isotopism problem for semifields with nuclei that are larger than its centre.

A graph is k -linked if any k disjoint vertex-pairs can be joined by k disjoint paths. We slightly improve a lower bound on the linkedness of polytopes. This results in exact values for the minimal linkedness of 7-, 10- and 13-dimensional polytopes. We analyze in detail linkedness of d -polytopes on at most (6 d + 7)/5 vertices. In that case, a tight lower bound on minimal linkedness is derived, and examples meeting this lower bound are constructed. These examples contain a class of examples due to Gallivan.

In this article we extend the proof given by Biswas and Gómez [Quart. J. Math. 54: 159–169, 2003] of a Torelli theorem for the moduli space of Higgs bundles with fixed determinant, to the parabolic situation.

In real space forms, given a regular hypersurface S , it is known that the integral over the space of s -planes of the mean curvature integral of intersections with S is a multiple of the mean curvature integral of S . We study the corresponding expression in a complex space form. The role of the s -planes is played by the complex s -planes. We prove that the same property does not hold but another term appears. We express this term as the integral of the normal curvature in the direction obtained from applying the complex structure to the normal direction. As an application, we give the measure of the set of complex lines meeting a compact domain in a complex space form, and we characterize the reproductive continuous invariant valuations of degree 2 n – 2 in the standard Hermitian space.

We prove that a Hilbert domain which admits a quasi-isometric embedding into a finite-dimensional normed vector space is actually a convex polytope.

In 1982 P. Cameron gave a characterisation of dual polar spaces of finite rank viewed as point-line spaces. This characterisation makes essential use of the fact that dual polar spaces of finite rank have finite diameter. Our goal is to give a characterisation which includes dual polar spaces of infinite rank. Since dual polar spaces of infinite rank are disconnected, we introduce a point-relation that denotes pairs of points at “maximal distance”, and we call this an opposition relation. This approach is in the spirit of the theory of twin buildings.

We apply the graph complex approach of [Markl, Differential Geom. Appl. 27: 257–278, 2009] to vector fields depending naturally on a set of vector fields and a linear symmetric connection. We characterize all possible systems of generators for such vector-field valued operators including the classical ones given by normal tensors and covariant derivatives. We also describe the size of the space of such operators and prove the existence of an ‘ideal’ basis consisting of operators with given leading terms which satisfy the (generalized) Bianchi–Ricci identities without the correction terms. The proofs given in this paper combine the classical methods of normal coordinates with the graph complex method.

We give a bound for the number of real solutions to systems of n polynomials in n variables, where the monomials appearing in different polynomials are distinct. This bound is smaller than the fewnomial bound if this structure of the polynomials is not taken into account.

For positive integers m and n , let S 1 , . . . , S n be nonempty sets, each having a subset of at most m labeled points, and let P be a nonempty set. Assume that a subset of each point p of P is associated with some (visibility) subset V ( p ) of S 1 ∪ ⋯ ∪ S n . We ask the following question, motivated by a result of Valentine: When will there exist a set of labeled points s 1 , . . . , s n , s i in S i , 1 ≤ i ≤ n , such that each V ( p ) set contains at least one of the s i points? Three results are given here. The first concerns the case for general S i sets. The second concerns the case in which the S i sets are simple arcs and V ( p ) ∩ S i is connected for each p and each i . The third result interprets the S i sets as compact, convex islands in the plane and interprets the V ( p ) sets as visible shorelines of S i ∪ ⋯ ∪ S n .

This paper shows the existence of right-angled buildings with arbitrary predetermined rational -Betti numbers.