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(OX(nD)) := dimC H0(X,OX(nD)). In the introduction to his collected works , 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 ), 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.
"Effective Behavior on Multiple Linear Systems." Asian J. Math. 8 (2) 287 - 304, April, 2004.