Maximal variation of linear systems

TL;DR

This paper investigates conditions under which linear systems on complex varieties exhibit maximal variation, meaning their elements vary in the largest possible moduli space, across various types of algebraic varieties.

Contribution

It identifies specific cases where linear systems are expected to have maximal variation, expanding understanding of moduli behavior in algebraic geometry.

Findings

Maximal variation occurs in hypersurfaces and double covers of projective space.

K3 surfaces and hyperkähler manifolds often exhibit maximal variation.

Abelian varieties can also have linear systems with maximal variation.

Abstract

Let X be a smooth projective complex variety, and L a line bundle on X . We say that the linear system |L| has maximal variation if its elements have the maximum number dim|L| of moduli. We discuss some cases where this situation is expected: hypersurfaces, double coverings of the projective space, K3 surfaces, hyperkahler manifolds, and abelian varieties.