Open Access
April, 2004 Effective Behavior on Multiple Linear Systems
Sheng-Li Tan
Asian J. Math. 8(2): 287-304 (April, 2004).


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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Sheng-Li Tan . "Effective Behavior on Multiple Linear Systems." Asian J. Math. 8 (2) 287 - 304, April, 2004.


Published: April, 2004
First available in Project Euclid: 24 June 2004

zbMATH: 1092.14009
MathSciNet: MR2129538

Rights: Copyright © 2004 International Press of Boston

Vol.8 • No. 2 • April, 2004
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