Manifolds Containing an Ample P¹-bundle

2016/02/01 by Daniel Litt, Litt, Daniel
Mathematics · #Algebraic Geometry (math.AG) #FOS: Mathematics #math.AG

paper · pdf · doi:10.48550/arxiv.1602.00716

4 pages; comments welcome!

arxiv created 2016/02/01 · arxiv updated 2016/02/03

Abstract

Sommese has conjectured a classification of smooth projective varieties X containing, as an ample divisor, a P^d-bundle Y over a smooth variety Z. This conjecture is known if d>1, if dim(X)<5, or if Z admits a finite morphism to an Abelian variety. We confirm the conjecture if the Picard rank rho(Z)=1, or if Z is not uniruled. In general we reduce the conjecture to a conjectural characterization of projective space: namely that if W is a smooth projective variety, E is an ample vector bundle on W, and Hom(E, T_W) is non-zero, then W is isomorphic to Pⁿ.

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