October 30, 2004 (Saturday)
Room
517, Meng Wah Complex, HKU
Professor Sheng-Li Tan
On the Riemann-Roch Problem
Abstract
Riemann-Roch
problem is one of the most important classical problems in algebraic geometry.
It is to compute effectively the dimension of a multiple linear system |nD| on a complex projective
manifold X for large n. If X is a curve, the problem was solved
completely by Riemann and Roch in the 1850s due to
their formula. However, if the dimension of X is at least 2, this problem is not
completely solved although we still have Riemann-Roch
formula. In the surface case, Zariski (1962) and Cutkosky –Srinivas (1993) gave an
ineffective solution to the problem, namely they proved the periodicity of the
dimension as a function of n for large n, and the
lower bound on n is ineffective.
In the special case where D is ample, it is well known that
the dimension can be computed by RR formula if n is bigger than some number N. The classical Effective Postulation
Problem is to find
an effective N. The
effective postulation problem for the linear systems of plane curves with some fixed
singularities at given points were studied by many famous mathematicians, e.g.,
Cayley, Castelnuvo,
Hilbert, Noether, Severi, Zariski, ….
In this talk, I am going to present optimal
effective solutions to these classical problems in the surface case. As an
application, we will give also some optimal lower bounds on Sheshadri’s
constants of line bundles on some surfaces.