DIVISION ALGEBRAS THAT RAMIFY ONLY ON A PLANE QUARTIC CURVE

Abstract. Let k be an algebraically closed field of characteristic 0. We prove that any division algebra over k(x, y) whose ramification locus lies on a quartic curve is cyclic. 1. Main result Let k be an algebraically closed field of characteristic 0, L = k(x,y) its purely transcendental extension of degree 2. We are interested in division L-algebras with ramification only at a quartic curve. Our main result (Theorem 1.2), which grew out from our earlier paper [11], shows that any such algebra is a symbol algebra whose index equals the exponent. This is a generalization of earlier results of Ford, Van den Bergh and Jacob on cyclicity of k(x,y)-algebras ramified either along a cubic curve, or a hyperelliptic curve, or a special quartic curve (see [5], [6], [7], [9], [16]). In contrast to the above cited papers (and to the paper of de Jong [10] who proved the equality of index and the exponent for any central simple algebra defined over an extension of k of transcendence degree 2, regardless of its ramification locus), our methods are quite elementary and describe the cyclic algebra explicitl