Volume 22, Issue 4

We consider weak Fano manifolds with small contractions obtained by blowing up successively curves and subvarieties of codimension 2 in the product of a projective space and a projective line. We give a classification result for a special case. We determine the nef cones and describe the extremal contractions for related smooth projective varieties.

We discuss criteria for a stable map of genus two and degree 4 to the projective plane to be smoothable, as an application of our modular desingularisation ofM‾2,nPr,dmain $\overline{\mathcal{M}}_{2, n}\left(\mathbb{P}^r, d\right)^{\text {main}}$via logarithmic geometry and Gorenstein singularities.

We prove the following result: Let K be a strictly convex body in the Euclidean space ℝ n , n ≥ 3, and let L be a hypersurface which is the image of an embedding of the sphere 𝕊 n –1 , such that K is contained in the interior of L . Suppose that, for every x ∈ L , there exists y ∈ L such that the support cones of K with apexes at x and y differ by a central symmetry. Then K and L are centrally symmetric and concentric.

The paper contributes to Majorana theory . Among the eight non-trivial Norton–Sakuma algebras, four algebras are closed on the set of Majorana generators. These algebras are 2 A , 2 B , 3 C and 4 B . The classification of Majorana representations restricted to the closed shapes was anticipated for a long time. In the present article the classification is achieved for shapes restricted to 2 A , 3 C and 4 B and for the set of generating involutions in the target group forming a single conjugacy class. Timmesfeld’s classification of {3, 4}+-transposition groups reduces to consideration of just three groups: L 3 (2), G 2 (2) ' and 3 D 4 (2). Each of these groups possesses a unique Majorana representation of the required shape. Only the representation of L 3 (2), known before, is based on an embedding into the Monster.

In this paper, Hopf hypersurfaces in a complex projective plane ℂ P 2 ( c ) or a complex hyperbolic plane ℂ H 2 ( c ) with constant scalar curvature are classified. For a non-Hopf hypersurface in ℂ P 2 ( c ) with constant scalar curvature r , it is proved that if the structure vector field is an eigenvector of the Ricci operator, then either r = 7 c /2 or r = 3 c /2. Moreover, these two cases are determined completely under an additional condition.

Let L be a full-rank lattice in ℝ d and write L + for the semigroup of all vectors with nonnegative coordinates in L . We call a basis X for L positive if it is contained in L + . There are infinitely many such bases, and each of them spans a conical semigroup S ( X ) consisting of all nonnegative integer linear combinations of the vectors of X . Such S ( X ) is a sub-semigroup of L + , and we investigate the distribution of the gaps of S ( X ) in L + , i.e. the points in L + ∖ S ( X ). We describe some basic properties and counting estimates for these gaps. Our main focus is on the restrictive successive minima of L + and of L + ∖ S ( X ), for which we produce bounds in the spirit of Minkowski’s successive minima theorem and its recent generalizations. We apply these results to obtain analogous bounds for the successive minima with respect to Weil heights of totally positive sub-semigroups of ideals in totally real number fields.

Masures are generalizations of Bruhat–Tits buildings. They were introduced by Gaussent and Rousseau in order to study Kac–Moody groups over valued fields. We prove that the intersection of two apartments of a masure is convex. Using this, we simplify the axiomatic definition of masures given by Rousseau.

We describe the development of the mathematics of Helmut R. Salzmann (3. 11. 1930 – 8. 3. 2022) and the main difficulties he was facing, documenting his lifelong productivity and his far reaching influence. We include a comprehensive bibliography of his work.

We discuss various aspects of isometric group actions on proper metric spaces. As one application, we show that a proper and Weyl transitive action on a euclidean building is strongly transitive on the maximal atlas (the complete apartment system) of the building.

Using the Klein correspondence, regular parallelisms of PG(3, ℝ) have been described by Betten and Riesinger in terms of a dual object, called a hyperflock determining ( hfd ) line set. In the special case where this set has a span of dimension 3, a second dualization leads to a more convenient object, called a generalized star of lines. Both constructions have later been simplified by the author. Here we refine our simplified approach in order to obtain similar results for regular parallelisms of oriented lines. As a consequence, we can demonstrate that for oriented parallelisms, as we call them, there are distinctly more possibilities than in the non-oriented case. The proofs require a thorough analysis of orientation in projective spaces (as manifolds and as lattices) and in projective planes and, in particular, in translation planes. This is used in order to handle continuous families of oriented regular spreads in terms of the Klein model of PG(3, ℝ). This turns out to be quite subtle. Even the definition of suitable classes of dual objects modeling oriented parallelisms is not so obvious.

Kleinewillinghöfer classified in 1979 automorphism groups of Laguerre planes with respect to linearly transitive subgroups of central automorphisms and obtained a multitude of types. All feasible Kleinewillinghöfer types of 2-dimensional Laguerre planes were completely determined in 2021. In this paper we investigate the Kleinewillinghöfer types of 4-dimensional Laguerre planes with respect to the automorphism groups of these planes and show that of the 49 types Kleinewillinghöfer described, only twelve are feasible in 4-dimensional Laguerre planes. Examples of four of these type are provided.

The present paper investigates 16-dimensional compact translation planes with automorphism groups of dimension d between 35 and 37; planes with groups of higher dimensions have been classified by Hähl. We obtain a complete classification for d = 37 (up to isomorphisms). It turns out that these planes have Lenz type V and are already described in a recent paper of Hähl and Meyer [10]. Moreover, we give a partial classification for d = 35 and d = 36. The latter case will be completely finished in a forthcoming paper [16] of the author, while the case where d = 35 is completed except for groups whose maximal compact subgroups are 9-dimensional.