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This book is about the smooth classification of a certain class of
algebraicsurfaces, namely regular elliptic surfaces of geometric
genus one, i.e. elliptic surfaces with b1 = 0 and b2+ = 3. The
authors give a complete classification of these surfaces up to
diffeomorphism. They achieve this result by partially computing one
of Donalson's polynomial invariants. The computation is carried out
using techniques from algebraic geometry. In these computations
both thebasic facts about the Donaldson invariants and the
relationship of the moduli space of ASD connections with the moduli
space of stable bundles are assumed known. Some familiarity with
the basic facts of the theory of moduliof sheaves and bundles on a
surface is also assumed. This work gives a good and fairly
comprehensive indication of how the methods of algebraic geometry
can be used to compute Donaldson invariants.
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