Irregular surfaces on hypersurfaces of degree 4 with non-degenerate isolated singularities (Q2157795)

The author works over the complex numbers; he studies hypersurfaces \(X\subset \mathbb P^4\). The case that \(X\) is contained in a quadratic or cubic surface \(Y\subset \mathbb P^4\) is settled by \textit{L. Roth} [Proc. Lond. Math. Soc. (2) 42, 142--170 (1936; JFM 62.0770.05)], and \textit{M. Masuda} and \textit{T. Petrie} [Proc. Natl. Acad. Sci. USA 88, No. 20, 9061--9064 (1991; Zbl 0753.14042)]. In the paper under review the author treats the case that \(X\) is a smooth irregular surface contained in a quartic hypersurface \(Y\subset \mathbb P^4\) and not in one of smaller degree. His main result is Theorem 1.1: If \(Y\) has at most ordinary double points, then \(X\) is a degree \(8\) elliptic conic bundle; such bundles were discovered by \textit{H. Abo} et al. [Math. Z. 229, No. 4, 725--741 (1998; Zbl 0954.14028)]. The case that \(X\) is a smooth scroll or a conic bundle is well known. In the general case, the author interprets the quartic threefold \(Y\) as a non-zero global section \(s\) of the twisted conormal bundle \(N^\vee_{X/\mathbb P^4}(4H)\) where \(H\) is a hyperplane section of \(X\), and studies the Koszul sequence associated to \(s\).