Construction of a birational isomorphism of a cubic threefold and Fano variety of the first kind with \(g=8\), associated with a normal rational curve of degree 4
It is well-known that a cubic threefold V is birationally isomorphic to a section of the Grassmannian G(2,6). In this paper another construction of a birational isomorphism is given. To describe it the author uses the notations: C: the generic rational normal curve on V of degree 4; \((S_ i)^{16}_{i=1}:\) the chords of C which lie on V; \(\sigma: V_ 1\to V:\) the blow up of C and all \(S_ i\); H: a hyperplane section of V. Theorem. (a) The linear system \(L=| 8H-5C-2\sum S_ i|\) on \(V_ 1\) is without base points and defines a birational isomorphism \(\phi\) of \(V_ 1\) and a section \(V^ 3_{14}\) of G(2,6) by a subspace of codimension 5; the map \(\phi\) contracts the surface \(D\in | 3H-2C- \sum S_ i| \quad to\) a point P and the inverse images \(S_ i'\) of the lines \(S_ i\) are contracted onto conics of \(V^ 3_{14}\); (b) the inverse map \(\phi^{-1}:\quad V^ 3_{14}\to V\) is given by the linear system \(| 2H-5P-\sum S_ i'|,\) where \(H'\) is a hyperplane section of \(V^ 3_{14}\); the map \(\phi^{-1}\) transforms the sixteen conics through P onto the chords of C; moreover, the surface \(D'=\phi (C)\in | 3H'-8P-2\sum S_ i'|\)\ is contracted onto C by \(\phi^{-1}\).
scientific article; zbMATH DE number 3964032