The Fine Moduli Space of Representations of Clifford Algebras

摘要

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 (αu+βv)d−f(α,β) with α and β 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 defined by the equation wd=f(u,v), and produces U_f,r as a quasiprojective variety in the moduli space of stable vector bundles over C with rank r and degree d_r=r(d+g−1), where g denotes the genus of C.