Volume 9, Issue 1

The present paper contains the construction of a family of 4-dimensional translation planes with a 7-dimensional collineation group. The planes of this family do not appear in the comprehensive classification [Betten, J. Reine Angew. Math. 285: 126–148, 1976, Betten, Geometriae Dedicata 3: 405–440, 1974/75, Betten, Math. Z. 154: 125–141, 1977]. We furthermore determine the isomorphism types of the new planes.

In this paper, we introduce and study several numbers and functions associated to a convex body, by them we can study the covering number, the blocking number and even Borsuk's partition number of a convex body in one setting.

Hiramine, Matsumoto and Oyama [Osaka J. Math. 24: 123–137, 1987] made the remarkable discovery that every translation plane of order q 2 that is 2-dimensional over its kernel produces translation planes of order q 4 . This construction is studied, with emphasis on isomorphisms among the resulting planes.

We give examples of characterizations of point shadows of apartments of buildings using Theorem 3.3 of [Kasikova, European J. Combin. 28: 1493–1529, 2007], and prove some properties of point-line truncations of J -Grassmann geometries of buildings, used in those examples, for all buildings and all choices of J .

We obtain 866 isomorphism classes of five-dimensional nonsingular toric Fano varieties using a computer program and the database of four-dimensional reflexive polytopes. The algorithm is based on the existence of facets of Fano polytopes having small integral distance from any vertex.

Some pseudo-Riemannian modifications of 6-dimensional and 7-dimensional Riemannian g.o. spaces are presented as pseudo-Riemannian homogeneous spaces with noncompact isotropy groups. These examples have the property that all geodesics are homogeneous up to a set of measure zero. Based on these examples, conjectures on geodesic graphs are formulated.

In this note we construct some series of translation planes of order q 2 which possess two commuting homology groups of order q – 1 and ( q + 1)( q – 1, 2) respectively. These translation planes can be considered as relatives of nearfield planes: for nearfield planes the nontrivial part of the associated spread set is a group whereas in our case this set is the product of two groups.

Here we study the α-stability for holomorphic triples on bielliptic curves. In particular some existence theorems for α-stable triples are proved using as main tool the study of holomorphic triples on elliptic curves. Elementary transformations of triples are also taken into consideration.

We improve Ottaviani's splitting criterion for vector bundles on a quadric hypersurface and obtain the equivalent of the result by Rao, Mohan Kumar and Peterson. Then we give the classification of rank 2 bundles without “inner” cohomology on 𝒬 n ( n > 3). It surprisingly exactly agrees with the classification by Ancona, Peternell and Wisniewski of rank 2 Fano bundles.