The extension of the reduced Clifford algebra and its Brauer class. (Q1416700)

For \(f\) a form of degree \(d\) in two variables over a field \(k\), the Clifford algebra \(C_f\) is the quotient of the free associative \(k\)-algebra \(k\{ x_1, x_2 \}\) modulo the ideal \(I\) generated by \[ \{ (\alpha_1x_1 + \alpha _2x_2)^d - f(\alpha _1, \alpha _2) \mid \alpha _1, \alpha _2 \in k \}. \] In [Trans. Am. Math. Soc. 355, 3181--3208 (2003; Zbl 1035.11015)] the author defined the reduced Clifford algebra \(A_f\), and showed that \(A_f\) is Azumaya over its center, hence represents a class in \(\text{Br} (U)\), the Brauer group of the affine ring of the complement \(U\) of a certain divisor in \(\text{Pic}_{C/k}^{d+g-1}\), where \(C\) is the curve defined by \(w^d = f(u, v)\) of genus \(g\). Here the author shows that the Brauer class of \(A_f\) extends to a class in \(\text{Br} (\text{Pic}_{C/k}^{d+g-1})\). If \(d\) is odd, so that \(\text{Pic}_{C/k}^{d+g-1}\) and the Jacobian \(J(C)\) are isomorphic over \(k\), then \(A_f\) in \(\text{Br}(U)\) is split iff \(J(C)\) has a \(k\)-rational point. This generalizes a result for \(d=3\) of \textit{D. E. Haile} [J. Algebra 146, 514--520 (1992; Zbl 0758.11022)].