On the Clifford algebra of a binary form

Kulkarni, Rajesh

doi:10.1090/S0002-9947-03-03293-8

On the Clifford algebra of a binary form
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by Rajesh S. Kulkarni PDF

Trans. Amer. Math. Soc. 355 (2003), 3181-3208 Request permission

Abstract:

The Clifford algebra $C_f$ of a binary form $f$ of degree $d$ is the $k$-algebra $k\{x, y\}/I$, where $I$ is the ideal generated by $\{(\alpha x + \beta y)^d - f(\alpha , \beta ) \mid \alpha , \beta \in k\}$. $C_f$ has a natural homomorphic image $A_f$ that is a rank $d^2$ Azumaya algebra over its center. We prove that the center is isomorphic to the coordinate ring of the complement of an explicit $\Theta$-divisor in $\operatorname {Pic}_{C/k}^{d + g - 1}$, where $C$ is the curve $(w^d - f(u, v))$ and $g$ is the genus of $C$.

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