# Alessio Corti's Flips for 3-folds and 4-folds PDF

By Alessio Corti

This edited selection of chapters, authored by way of major specialists, presents an entire and primarily self-contained development 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 evidence 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 resources for Flips for 3-folds and 4-folds**

**Sample text**

I deﬁne the restriction D0 = resS D of D to S as follows. Pick a model f : Y → X such that D = DY ; let S ⊂ Y be the proper transform. I deﬁne resS D = DY |S , where DY |S is the ordinary restriction of divisors. ) It is easy to see that the restriction does not depend on the choice of the model Y → X . 40 Restriction is additive and monotone, in the sense that (1) (D1 + D2 )0 = D01 + D02 , and (2) if D1 ≥ D2 , then D01 ≥ D02 . 41 The restriction of a mobile b-divisor is a mobile b-divisor. 42 Restriction of mobile b-divisors is compatible with restriction of rational functions in the following sense.

Section 4 contains complete details of the construction of 3-fold pl ﬂips. It opens with the proof that the ﬁnite generation conjecture implies the existence of pl ﬂips, and ends with a proof of the ﬁnite generation conjecture in dimension 2, using the techniques developed in the previous section. 3 Other surveys Shokurov’s ideas on ﬂips are published in [Sho03], see also the other papers in the same volume; in particular, the 3-fold case is surveyed in [Isk03]. 1 Summary The ﬁrst goal of this section is to give the deﬁnition of two ﬂavours of log terminal singularities: klt and plt.

This is just as well, since OX A(X , D) = OX if and only if the pair (X , D) has klt singularities. 1], see also Chapter 5. Proof We may assume that X is afﬁne. Choose a log resolution f : Y → X , and write KY = f ∗ (KX + D) + ai Ai , where the support A = ∪Ai is a simple normal crossing divisor. Note that, by deﬁnition, AY = ai Ai . 14, if g : Y → Y is another model of X , then AY = g ∗ AY + (effective & exceptional). This shows that f∗ OY AY = f∗ OY AY (where f = fg), from which it follows ai Ai is a coherent sheaf.