Compactifications of \({\mathbb{C}}^ 3\). II (Q1119708)

[For part I of this paper see the author and \textit{M. Schneider}, Math. Ann. 280, No.1, 129-146 (1988; Zbl 0651.14025).] This paper shows that Fano compactifications of \({\mathbb{C}}^ 3\) with \(b_ 2=1\) and index \(r=1\) have genus \(g(:=+(c^ 3_ 1/2)+1)=12\), and are thus rational with \(b_ 3=0\). The proof rests on the classification of Fano threefolds of Iskovskij-Shokurov. Such a compactification exists, as shown by M. Furushima. Fano compactifications with \(b_ 2=1\) and (4\(\geq)r\geq 2\) are classified in part I of this paper. Other papers with \textit{S. Kosarew} show that compactifications of \({\mathbb{C}}^ 3\) with \(b_ 2=1\) are Moishezon.