Abstract
AbstractIn this paper, we provide explicit generators for the Picard groups of cyclic Brauer-Severi varieties defined over the base field. In particular,we provide such generators for all Brauer-Severi surfaces. To produce these generators we use the theory of twists of smooth plane curves.
Highlights
Let B k be a Brauer-Severi variety over a perfect eld k, that is, a projective variety of dimension n isomorphic over k to Pnk
In this paper, we provide explicit generators for the Picard groups of cyclic Brauer-Severi varieties de ned over the base eld
We provide such generators for all Brauer-Severi surfaces. To produce these generators we use the theory of twists of smooth plane curves
Summary
Let B k be a Brauer-Severi variety over a perfect eld k, that is, a projective variety of dimension n isomorphic over k to Pnk. An algorithm to compute these equations for any Brauer-Severi variety is given in [9]. We show an explicit concrete generator of the Picard group of any Brauer-Severi variety corresponding to a cyclic algebra in its class inside the Brauer group Br(k) of k. For a BrauerSeveri surface B and any integer r ≥ , we obtain a generator for r Pic(B) from twists of a Fermat type smooth plane curve, see Theorem 4.2. Let B be the Brauer-Severi surface corresponding to a cyclic algebra (L k, σ, a) of dimension as in Theorem 2.5. A smooth model of B inside Pk is given by the intersection ∩τ∈Gal(L k) τ X where X L is the variety in PL de ned by the set of equations:.
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