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We define generalized Clifford parallelisms in PG(3, F) with the help of a quaternion skew field H over a field F of arbitrary characteristic. Moreover we give a geometric description of such parallelisms involving hyperbolic quadrics in projective spaces over suitable quadratic extensions of F .

In this paper we focus on the description of the automorphism group Γ of a Clifford-like parallelism on a 3-dimensional projective double space P(H F ), ℓ , r over a quaternion skew field H (with centre a field F of any characteristic). We compare Γ with the automorphism group Γ ℓ of the left parallelism ℓ , which is strictly related to Aut(H). We build up and discuss several examples showing that over certain quaternion skew fields it is possible to choose in such a way that Γ is either properly contained in Γ ℓ or coincides with Γ ℓ even though ℓ .Mathematics Subject Classification (2010): 51A15, 51J15

Here we study the α-stability for holomorphic triples over curves of genus g = 1. We provide necessary and sufficient conditions for the moduli space of α-stable triples to be non-empy and, in these cases, we show that it is smooth and irreducible.