By Herbert Clemens, Janos Kollár

This quantity collects a sequence of survey articles on complicated algebraic geometry, which within the early Nineteen Nineties used to be present process an enormous swap. Algebraic geometry has spread out to rules and connections from different fields that experience characteristically been distant. This ebook offers a good suggestion of the highbrow content material of the swap of course and branching out witnessed through algebraic geometry some time past few years.

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**Example text**

Then C can be flipped or flopped on Z over W . A sequence of flips on Z/W (possibly preceded by a single flop), followed by a divisorial contraction, gives the original flip of Xi → W . 2). The flop at the beginning sometimes occurs, but in this situation it can be easily constructed directly, so no previous knowledge of (terminal) flops is assumed here. This point deserves to be emphasized a little more. The reader of [Ka] may notice that, as a byproduct of the above mentioned calculation leading to the inequality KZ · C ≤ 0, it is easy to construct a divisor B in a neighbourhood of C ⊂ X such that KX + B is log terminal and numerically trivial on C (we do not even need h = 1 for this).

For D general, the D-minimal model program should be considered as a way to obtain some kind of “Zariski decomposition” of D. In particular, it is not clear (certainly not to me) a priori that there should be any interesting reason at all to consider general logarithmic divisors KX + B, where B = bi Bi is allowed to have rational coefficients bi , 0 < bi ≤ 1. Nevertheless, these divisors have been profitably used (especially by Kawamata) since the earlier days of the 40 ALESSIO CORTI theory, to direct or construct portions of the (genuine) Mori program, especially in relation to flops and flips.

Hint: show that such a bundle has always a nowhere vanishing section, and use induction on the rank). We choose a small disk D ⊂ X around p (actually, to avoid convergence problems we take D = Spec (O), where O is the completed local ring of X at p, but this makes essentially no difference). We then consider triples (E, ρ, σ), where E is a vector bundle on X, ρ an algebraic trivialization of E over X p and σ a trivialization of E over D. Over D p these two trivializations differ by a holomorphic map D p −→ GLr (C) that is meromorphic at p, that is, given by a Laurent series γ ∈ GLr C((z)) .