By Herbert Clemens, Janos Kollár

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

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There are by now several available proofs of these results. The fact that the k dimension of BX (r) is given by the Verlinde formula follows from the work of k Tsuchiya, Ueno and Yamada [T-U-Y]. They show that the dimension of BX (r) is independent of the curve X, even if X is allowed to have double points. Then k it is not too difficult to express BX (r) in terms of analogous spaces for the normalization of X (this is called the factorization rules by the physicists). One is thus reduced to the genus-0 case (with marked points), that is, to a problem in the theory of representations of semisimple Lie algebras, which is non-trivial in general (actually I know no proof for the case of an arbitrary semisimple Lie algebra), but rather easy for the case of slr (C).

Log abundance for ν(K + B) = dim(X) is an immediate consequence of the base point free theorem. The proof of the (genuine) abundance theorem for threefolds (mainly due to Kawamata and Miyaoka) is very long and complicated: it can be found in [Ko], together with proper attributions. Roughly speaking, the proof is divided in two parts, requiring entirely different techniques. Here is a quick summary: A) First one shows that |mK| = ? for some m. This is quite hard, and I refer to [Ko, Ch. 9] for an extremely attractive presentation.

A) The case k = 1 is exactly Theorem 2. b) The two spaces have the same dimension. ) because the map τ1,k : SU X (k) × J g−1 (X) −→ UX (k) ∗ ∗ is an ´etale (Galois) covering of degree k 2g , and τ1,k (M) ∼ pr L ⊗ = 1 ∗ pr2 OJ (kΘ). Therefore we get 1 χ(Lr ) χ(OJ (krΘ)) k 2g rg = g dim H 0 ((SU X (k), Lr ) . k ∗ (k), Mr ) = χ(Mr ) = dim H 0 ((UX Now, Theorem 5 shows that k −g dim H 0 ((SU X (k), Lr ) is symmetric in k and r, which proves our assertion. c) Therefore it is enough to prove, for example, the surjectivity of the map ϑk,r , which has the following geometric meaning: Question 6 .