The unirationality of all conic bundles implies the unirationality of the quartic threefold. -- Appendix by Mariana Marchisio: Every quartic 3-fold contains a rational surface (Q2717167)

A conic bundle is a 3-dimensional algebraic variety \(X\), defined over any field \(K\), together with a morphism \(\pi : X \rightarrow S\), where \(S\) is a rational surface (e.g.\ \(S = \mathbb P^2\)), such that the fibres of \(\pi\) are conics. An algebraic variety \(Y\) is unirational if there exists a rational dominant map \(\chi: \mathbb P^r \rightarrow Y\). The author considers two important open questions: NEWLINENEWLINENEWLINE(A) is every conic bundle unirational? (Fano conjectured the answer to be no, but no proof has been given.) NEWLINENEWLINENEWLINE(B) Is the generic quartic threefold unirational? NEWLINENEWLINENEWLINEThe author proves that a positive answer to (A) implies a positive answer to (B). He uses a result of M. Marchisio, proved in an appendix to the paper, which says that every quartic 3-fold \(V \subset \mathbb P^4\) contains a rational surface, which can even be chosen passing through an arbitrary point of \(V\).NEWLINENEWLINEFor the entire collection see [Zbl 0960.00042].