Effective Behavior on Multiple Linear Systems

Abstract

It is a fundamental problem in algebraic geometry to understand the behavior of a multiple linear system |nD| on a projective complex manifold X for large n. For example, the well-known Riemann-Roch problem is to compute the function n ↦ h⁰(O_X(nD)) := dim_C H⁰(X,O_X(nD)). In the introduction to his collected works [33], Zariski cited the Riemann-Roch problem as one of the four "difficult unsolved questions concerning projective varieties (even algebraic surfaces)". The other natural problems about |nD| are to find the fixed part and base points (see [32]), the very ampleness, the properties of the associated rational map and its image variety, the finite generation of the ring of sections.

For a genus g curve X, Riemann-Roch theorem gives good and effective solutions to these problems.