Volume 13, Issue 3

In this note we investigate the (E)-index and the (E)-nullity of vacuum solutions from the two-torus into the two sphere.

We classify Fano threefolds with only terminal singularities whose canonical class is Cartier and divisible by 2 with the additional assumption that the G-invariant part of the Weil divisor class group is of rank 1 with respect to an action of some group G. In particular, we find a lot of examples of Fano 3-folds with “many” symmetries.

We classify Fano threefolds with only Gorenstein terminal singularities and Picard number greater than 1, satisfying the additional assumption that the G-invariant part of the Weil divisor class group is of rank 1 with respect to an action of some group G.

Suppose that Δ;Δʹ are two buildings each arising from a semisimple algebraic group over a field, a topological field in the former case, and that for both the buildings the Coxeter diagram has no isolated nodes. We give conditions under which a partially defined injective chamber map, whose domain is the subcomplex of Δ generated by a nonempty open set of chambers, and whose codomain is Δʹ, is guaranteed to extend to a unique injective chamber map. Related to this result is a local version of the Borel-Tits theorem on abstract homomorphisms of simple algebraic groups.

Let Π be a thick polar space of rank n ≥ 3. Pick a hyperplane F of Π and Η of Π. Define the elements of a biaffine polar space Γ to be those elements of Π which are not contained in F, or dually in Η. We show that Γ is a simply connected geometry, except for a few small exceptions for Π. We give a construction that leads to flag-transitive examples.

We study non-vanishing and birationality of big pluricanonical adjoint linear systems on a variety of general type X. In particular given L big and with sufficiently large volume, if X is a surface we prove that h0(2KX + L) 6= 0 and that j3KX + 2Lj gives a birational map; if X is a threefold we prove that h0(6KX + 5L) 6= 0 and that j8KX + 7Lj gives a birational map. We find analogous explicit results also for fourfolds of general type.

We classify hypersurfaces of revolution for which the principal curvatures are proportional. These hypersurfaces form a one-parameter family indexed by the reals. The subsequent classification unites at least two important families of hypersurfaces of revolution: the generalized catenoids and the equizonal ovaloids.

Metrical completeness for Bruhat-Tits buildings is a natural and useful condition. In this paper we determine which Bruhat-Tits buildings are metrically complete up to certain cases involving infinite-dimensionality and residue characteristic 2.

Results are proved indicating that the Veronese map vd often increases independence of both sets of points and sets of subspaces. For example, any d + 1 Veronesean points of degree d are independent. Similarly, the dth power map on the space of linear forms of a polynomial algebra also often increases independence of both sets of points and sets of subspaces. These ideas produce d + 1-independent families of subspaces in a natural manner

We classify those polarities of Moufang planes in characteristic two that have at least one absolute point. Along the way, we determine the Baer involutions of the planes in question and show that the corresponding Baer subplanes will be pappian or non-desarguesian Moufang planes.

If D ⊂ X is a curve with multiple points in a surface, a parabolic bundle defined on (X;D) away from the singularities can be extended in several ways to a parabolic bundle on a resolution of singularities. We investigate the possible parabolic Chern classes for these extensions.