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# Asian Journal of Mathematics

## Volume 8 (2004)

### Number 2

### Effective Behavior on Multiple Linear Systems

Pages: 287 – 304

DOI: http://dx.doi.org/10.4310/AJM.2004.v8.n2.a5

#### Author

#### Abstract

It is a fundamental problem in algebraic geometry to understand the behavior of a multiple linear system |*n*D| on a projective complex manifold X for large *n*. For example, the well-known Riemann-Roch problem is to compute the function *n* ↦ h^{0}(O_{X}(*n*D)) := dim_{C} H^{0}(X,O_{X}(*n*D)). 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 |*n*D| 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.