Nonbirational centers of linear projections of scrolls over curves

Abstract

We work over an algebraically closed field of characteristic zero. A nonbirational center of a projective variety is a point from which the variety is projected nonbirationally onto its image, whose locus plays an important role in a study of the double-point divisors and the defining equations of the variety. The purpose of this paper is to show that a scroll over a curve with some conditions has no nonbirational centers. Consequently such a nondegenerate scroll is cutting out by hypersurfaces of degree $d-e + 1$ for its degree $d$ and codimension $e$ in the projective space. On the other hand, examples of scrolls over curves with nonbirational centers are constructed.