Variations of the Albanese Morphism

We estimate the number of moduli of an n-dimensional variety X through the variation of its Albanese morphism. Refining upon our methods, we work out the classical Castelnuovo bound concerning the number m of moduli of irregular surfaces with birational Albanese map. We interpret our variation by means of higher Abel-Jacobi mappings theory and under the only hypothesis that X has a generically finite morphism to an Abelian variety A, we can bound from below the geometrical genus pg(X) in terms of the dimensions of A and X. Using the same framework, we characterize the hyperelliptic locus in Mg as the only close subvariety ℋ inside the moduli space of curves with dim ℋ ≥ 2g - 1 and torsion Abel-Jacobi image of the Ceresa cycle at its generic point.