Duality for elliptic normal surface scrolls

Abstract

For a variety X⊂PN, its mth order dual variety is the variety X∨m⊂(PN)∨ consisting of hyperplanes containing an mth osculating space of X. The strict dual variety X∗ is the smallest nonempty X∨m. All these dual varieties were described by Piene and G. Sacchiero [\cit MR0738534 (86c:14042) \endcit Comm. Algebra 12 (1984), no. 9-10, 1041–1066; MR0738534 (86c:14042)] in case X is a rational normal scroll. The main contribution of this paper is the description (dimension and degree) of the strict dual varieties of elliptic normal surface scrolls. First, by local computations, they describe all dual varieties for any decomposable normal scroll in terms of the dual varieties of the curves attached to the decomposition of the bundle. This solves the problem when the elliptic normal surface scroll is decomposable. For the nondecomposable case, they observe that such an X can be obtained as a projection of a decomposable elliptic normal surface scroll X′⊂PN+1 from a point of X′. From this observation and their former description they obtain their result in this case.