Division algebras that ramify only on a plane quartic curve with simply connected components. (Q1421735)

The author proved [\textit{T. J. Ford}, New York J. Math. 1, 178-183, electronic only (1995; Zbl 0886.16017)] that if \(\Lambda\) is a central division algebra over an algebraically closed field \(K\) of characteristic \(0\) such that the ramification divisor \(D\) is a singular plane cubic curve, then \(\Lambda\) is a symbol algebra of exponent equal to its index. His object here is to prove similar results when the ramification divisor of \(\Lambda\) is a quartic curve \(D\) whose irreducible components are all simply connected. These restrictions imply that each irreducible component of \(D\) is a rational curve and is either non-singular (and so isomorphic to the projective line) or singular with at most one cuspidal double point. The main theorem states: Let \(K\) be the rational function field of the projective plane \(\mathbb{P}^2\) and let \(\Lambda\) be a central division algebra over \(K\) of exponent \(n\), whose ramification divisor on \(\mathbb{P}^2\) has support in the curve \(D\). Assume that \(D\) has degree four and is the union of simply connected curves. Then \(\Lambda\) is a symbol algebra of index \(n\) and there is a cyclic Galois extension \(F= K(f^{1/n})\) such that the sequence \(0\to{_nB}(R)\to B(R)\to B(F)\) is exact, where \(B\) denotes the Brauer group and \(R\) is the affine coordinate ring of \(\mathbb{P}^2-D\). -- The proof is an analysis of cases, examining the possible irreducible components of \(D\) and their intersection configurations, and is based on the methods of the previous paper and results of \textit{T. J. Ford} [J. Algebra 147, No. 2, 365-378 (1992; Zbl 0791.13001)].