Projective varieties with nonbirational linear projections and applications

We work over an algebraically closed field of characteristic zero. The purpose of this paper is to characterize a nondegenerate projective variety X X with a linear projection which induces a nonbirational map to its image. As an application, for smooth X X of degree d d and codimension e e , we prove the “semiampleness” of the ( d − e + 1 ) (d-e+1) th twist of the ideal sheaf. This improves a linear bound of the regularity of smooth projective varieties by Bayer–Mumford–Bertram–Ein–Lazarsfeld, and gives an asymptotic regularity bound.