I will present two new properties of continuous linear systems on irregular varieties: the eventual degree  and a real extension of the continuous rank. These two constructions, of independent theoretical interest, allows us to extend to higher dimensions the classical results of Castelnuovo on the rank of multiplication maps on curves. Also, I will show how to obtain new higher dimensional Clifford-Severi inequalities in a short and conceptual way. Finally  I will explain how to classify varieties and line bundles on the Severi lines. This is a joint work with R. Pardini and L. Stoppino.

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