In this talk, we study the maximal variation problem for linear systems associated with a very ample line bundle, using Hodge theory and Picard–Lefschetz theory. We provide an affirmative answer to the maximal variation problem for a broad class of smooth projective varieties.

This includes varieties X of dimension n at least 2 with $p_g=h^{n,0}(X)>0$ and $H^{n-1,0}(X)=\{0\}$, Enriques surfaces, irregular surfaces with maximal Albanese dimension, smooth hyperkähler varieties, and all the smooth not Fano hypersurfaces in $\mathbb{P}^n$.

As a consequence, by a result of Beauville, we establish a Lefschetz property for the Jacobian rings of smooth hypersurfaces in $\mathbb{P}^n$ of degree at least n+1. This is based on a joint work in collaboration with Gian Pietro Pirola.