Division algebras that ramify only on a plane nodal cubic curve plus a line. (Q1428100)

Let \(k\) be an algebraically closed field of characteristic zero, and \(K\) the field of rational functions on the projective plane \(\mathbb{P}[x,y,z]:=\mathbb{P}^2\), where \(x\), \(y\) and \(z\) are commuting indeterminates over \(k\). It is known that then each central division \(K\)-algebra \(\Lambda\) ramifies along a divisor \(D=D_\Lambda\) on \(\mathbb{P}^2\). The paper under review shows that \(\Lambda\) is a symbol \(K\)-algebra of index equal to its exponent, i.e. to the order of the similarity class \([\Lambda]\) in the Brauer group of \(K\), provided that \(D\) is a reducible quartic curve and each irreducible component of \(D\) is birational to the projective line \(\mathbb{P}^1\). The proof of this result in the special case where the irreducible components of \(D\) are simply connected is published in another paper by the author [in Algebr. Represent. Theory 6, No. 5, 501-514 (2003; see the review Zbl 1044.16010 above)]. In the present paper, he considers the remaining case where \(D\) factors into an irreducible nodal cubic plus a line. The proof is obtained by analyzing separately the situation in which the curves are in general position and the alternative one.