On the Clifford algebra of a binary form

Let f be a homogeneous polynomial of degree d in two variables over a field k. The Clifford algebra of f, denoted by C f, is the k-algebra k{lcub} x,y{rcub}/I, where I is the ideal generated by &cubl0;ax+by d-fa,b &vbm0;a,b∈ k&cubr0; . When d = 3, Cf is an Azumaya algebra with center the affine coordinate ring of an elliptic curve. This elliptic curve is the Jacobian of the curve given by the equation w3-fu,v . We generalize this result to forms of degree d for arbitrary d. When d > 3, the Clifford algebra is itself not Azumaya over its center. However it has a natural homomorphic image, called the reduced Clifford algebra, which is an Azumaya algebra. We compute the center of the reduced Clifford algebra. We prove that, under certain hypothesis, the center is isomorphic to the coordinate ring of an open set in Picd+g-1 C/k . Here C is the curve given by the equation wd-fu,v and g is the genus of the curve C. In fact, the open set in the statement is the complement of an explicit Q -divisor in Picd+g-1 C/k .