LINEAR SYSTEMS on IRREGULAR VARIETIES

Let be a normal complex projective variety, a subvariety of dimension (possibly) and a morphism to an abelian variety such that injects into; let be a line bundle on and a general element.We introduce two new ingredients for the study of linear systems on. First of all, we show the existence of a factorization of the map, called the eventual map of on, which controls the behavior of the linear systems, asymptotically with respect to the pullbacks to the connected étale covers induced by the -th multiplication map of.Second, we define the so-called continuous rank function, where is the pullback of an ample divisor of. This function extends to a continuous function of, which is differentiable except possibly at countably many points; when we compute the left derivative explicitly.As an application, we give quick short proofs of a wide range of new Clifford-Severi inequalities, i.e., geographical bounds of the form where depends on several geometrical properties of, or.