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The Fine Moduli Space of Representations of Clifford Algebras

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2011-01-01
Given a fixed binary form f(u,v) of degree d over a field k, the associated Clifford algebra is the k-algebra C(f)=k{u,v}/I, where I is the two-sided ideal generated by elements of the form (alpha u+beta v)(d)-f(alpha,beta) with alpha and beta arbitrary elements in k. All representations of C(f) have dimensions that are multiples of d, and occur in families. In this article, we construct fine moduli spaces U=U(f,r) for the irreducible rd-dimensional representations of C(f) for each r >= 2. Our construction starts with the projective curve C subset of P(k)(2) defined by the equation w(d) = f(u, v), and produces U(f,r) as a quasiprojective variety in the moduli space M(r, d(r))of stable vector bundles over C with rank r and degree d(r)=r(d+g-1), where g denotes the genus of C.