Zariski decomposition of curves on algebraic varieties

Brian Lehmann, Jian Xiao

Published 2015-07-15Version 1

We introduce a Zariski decomposition for curve classes and use it to develop the theory of the volume function for curves defined by the second named author. For toric varieties and for hyperk\"ahler manifolds the Zariski decomposition admits an interesting geometric interpretation. With the decomposition, we prove some fundamental positivity results for curve classes, such as a Morse-type inequality. We compare the volume of a curve class with its mobility, yielding some surprising results about asymptotic point counts. Finally, we give a number of applications to birational geometry, including a refined structure theorem for the movable cone of curves.