Asian Journal of Mathematics

Volume 8 (2004)

Number 2

Effective Behavior on Multiple Linear Systems

Pages: 287 – 304



Sheng-Li Tan


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 [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.

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