Volume 15, Issue 2

Let D be a division algebra with center F and degree d > 2. Let K|F be any splitting field.We analyze the action of D × and SL 1 (D) on the spherical and affine buildings that may be associated to GL d (K) and SL d (K), and in particular show it is never strongly transitive. In the affine case we find examples where the action is nonetheless Weyl transitive. This extends results of Abramenko and Brown concerning the d = 2 case, where strong transitivity is in fact possible. Our approach produces some explicit constructions, and we find that for d > 2 the failure of the action to be strongly transitive is quite dramatic.

We study the syzygetic structure of projections of del Pezzo surfaces in order to construct singular Calabi-Yau threefolds. By smoothing those threefolds,we obtain new examples of Calabi-Yau threefolds with Picard group of rank 1. We also give an example of type II primitive contraction whose exceptional divisor is the blow-up of the projective plane at a point.

Gradient Ricci almost solitons were introduced by Pigola, Rigoli, Rimoldi and Setti [20]. They are defined as solitons except that the metric coefficient is allowed to be a smooth function rather than a constant. It is shown that any almost soliton is conformal to another almost soliton having a soliton function which is the negative of the original one. Uniqueness, and the case where both the source and target are solitons, are studied. Completeness of the target metric is also examined in the casewhere the source is Kähler and admits a special Kähler-Ricci potential in the sense of [10; 11].

We prove that a compact complex manifold endowed with a Kähler-Ricci soliton cannot be isometrically embedded in a complex projective space ℂℙ n in such a way that the Gauss map is rational, unless the metric is Einstein. This applies to hypersurfaces of complex compact homogeneous spaces canonically embedded in ℂℙ n .We moreover obtain two curvature constraints for invariant Kähler-Ricci solitons on complex manifolds acted on by a compact Lie group with cohomogeneity one.

Given a parameterized family of polynomial equations, a fundamental question is to determine upper and lower bounds on the number of real solutions a member of this family can have and, if possible, compute where the bounds are sharp. A computational approach to this problem was developed by Dietmaier in 1998 who used a local linearization procedure to move in the parameter space to change the number of real solutions. He used this approach to show that there exists a Stewart-Gough platform that attains the maximum of forty real assembly modes. Due to the necessary ill-conditioning near the discriminant locus, we propose replacing the local linearization near the discriminant locus with a homotopy-based method derived from the method of gradient descent arising in optimization. This new hybrid approach is then used to develop a new result in real enumerative geometry.

The concept of division pairs, a non-abelian version of Jordan division pairs, is introduced and a categorical equivalence between division pairs and Moufang sets is established. This is used to explain the non-uniqueness occurring in the description of Moufang sets in terms of pairs (U, τ) initiated by De Medts and Weiss.

Given a real curve, we study special linear systems called “very special” for which the dimension does not satisfy a Clifford type inequality.We classify all these very special linear systems when the gonality of the curve is small.

We investigate the intersection between the generalized quadrangle arising from a Hermitian surface H(3, q 2 ) and an elliptic quadric Q − (3, q 2 ) of PG(3, q 2 ). In odd characteristic we determine the possible intersection sizes between H(3, q 2 ) and Q − (3, q 2 ) under the hypothesis that they share the same tangent plane at a common point. When the characteristic is even, we determine the configuration arising from the intersection of H(3, q 2 ) and Q − (3, q 2 ), provided that the generators of H(3, q 2 ) that are tangents with respect to Q − (3, q 2 ) are the extended lines of a symplectic generalized quadrangle W(3, q) embedded in H(3, q 2 ). As a by-product, new infinite families of hyperovals on H(3, q 2 ) are constructed.

We study the geometry of semitube domains in ℂ 2 , in particular we extend the result of Burgués and Dwilewicz for semitube domains by dropping the smoothness assumption. We also prove various properties of non-smooth pseudoconvex semitube domains, obtaining a relation between pseudoconvexity of a semitube domain and the number of components of its vertical slices. Finally, we present an example of a non-convex domain in ℂ n such that its image under arbitrary isometries is pseudoconvex.

Linearly normal complex projective threefolds of degree 12, embedded in ℙ 6 , are classified in the framework of adjunction theory: a maximal list of the possible invariants is provided and many examples are constructed.