The extension of the reduced Clifford algebra and its Brauer class

The shape Clifford algebraCf of a binary form f of degree d is the k-algebra k{x,y}/I, where I is the ideal generated by {(αx+βy)d−f(α,β)∥α,βk}. Cf has a natural homomorphic image Af, called the reduced Clifford algebra, which is a rank d2 Azumaya algebra over its center. The center is isomorphic to the coordinate ring of the complement of an explicit Θ -divisor in PicC/kd +g −1, where C is the curve (wd−f(u,v)) and g is the genus of C ([9]). We show that the Brauer class of Af can be extended to a class in the Brauer group of PicC/kd + g −1. We also show that if d is odd, then the algebra Af is split if and only if the principal homogeneous space PicC/k1 of the jacobian of C has a k-rational point.