By Alessio Corti

This edited number of chapters, authored via prime specialists, offers a whole and basically self-contained building of 3-fold and 4-fold klt flips. a wide a part of the textual content is a digest of Shokurov's paintings within the box and a concise, entire and pedagogical facts of the lifestyles of 3-fold flips is gifted. The textual content incorporates a ten web page word list and is obtainable to scholars and researchers in algebraic geometry.

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**Extra info for Flips for 3-folds and 4-folds**

**Example text**

We are interested in comparing Mob MY |S + AS0 with MY |S . The punchline is this: by the vanishing theorem of Kawamata and Viehweg, H 1 Y , MY + AY = H 1 Y , KY + −f ∗ (KX + S + B) + MY = (0) (note that here we are using our assumption that the pair (X , S + B) is weak Fano); therefore, the natural restriction map H 0 Y , MY + AY → H 0 S , MY |S + AS0 is surjective. In other words, | MY |S +AS0 | contains no new sections, and the statement follows. 28 that saturation holds on models S → S higher than S .

36, M• is subadditive and D• is convex. Denote by H 0 (X , iM ) = R(X , D• ) R = R(X , M ) = i the associated pbd-algebra. Denote by R0 = Recall that, by deﬁnition, R0i = resS R the restricted algebra. R0i = Im res : H 0 (X , iM ) → k(S) . 40 M•0 is subadditive and D0• is convex). 55 starting with V = R0 . In particular, R0 ⊂ RS is an integral extension and R0 is ﬁnitely generated if and only if RS is ﬁnitely generated. 55. 3 RS is a Shokurov algebra. 4 The ﬁnite generation conjecture in dimension n − 1 implies existence of pl ﬂips in dimension n.

Roughly speaking, R0 = i R0i where R0i = Im H 0 X , OX (i(K + S + B)) → H 0 S, OS (i(KS + Diff S B)) . 6 below, that R is ﬁnitely generated if R0 is. Pbd-algebras The restriction of |i(K + S + B)| to S is not (a priori) a complete linear system; instead, it is a linear system with base conditions. To keep track of the base conditions, Shokurov develops the language of b-divisors. In the language of b-divisors, the restricted algebra R0 (or rather, more precisely, an algebra integral over it) is a pbd-algebra (pseudo-b-divisorial algebra).