A geometric construction of elliptic conic bundles in \(\mathbb{P}^4\) (Q1298006)

It is known that smooth conic bundles in \({\mathbb{P}}^4\) are either rational or elliptic [\textit{P. Ellia} and \textit{G. Sacchiero} in: Algebraic geometry, Pap. EURO-PROJ Conf., Catania 1993, Barcelona 1994, Lect. Notes Pure Appl. Math. 200, 49-62 (1998; Zbl 0940.14025) and \textit{R. Braun} and \textit{K. Ranestad}, ibid., 331-339 (1998)]. The elliptic conic bundles in \({\mathbb{P}}^4\) were first constructed by \textit{H. Abo, W. Decker} and \textit{V. Sasakura} [``An elliptic conic bundle in \(\mathbb{P}^4\) arising from a stable rank-3 vector bundle'', Math. Z. 229, No. 4, 725-741 (1998)] as degeneracy loci of four sections of a fixed rank five vector bundle in \({\mathbb{P}}^4\). In this paper the author gives a construction of a general elliptic conic bundle in \({\mathbb{P}}^4\) using Cremona transformations. For any smooth elliptic quintic scroll \(S_5 \subset {\mathbb{P}}^4\) he defines a Cremona transformation of \({\mathbb{P}}^4\) such that the image of \(S_5\) under this transformation is a smooth conic bundle. He proves that a general elliptic conic bundle \(S\) is obtained from a smooth quintic elliptic scroll by a Cremona transformation. This is done by constructing an inverse Cremona transformation from \(S\) to its minimal model.