By Alessio Corti
This edited number of chapters, authored by way of top specialists, offers an entire 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 evidence of the lifestyles of 3-fold flips is gifted. The textual content features a ten web page thesaurus and is offered to scholars and researchers in algebraic geometry.
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Extra resources for Flips for 3-folds and 4-folds
Shokurov’s proof in the 4-fold case is very convoluted and complicated; however, it rests on a rather appealing proof of existence of 3-fold pl ﬂips. The purpose of this chapter is to explain this construction. 2 Summary of the chapter This chapter is divided into three long sections. Section 2 is an introduction to log terminal singularities, the ﬂipping problem, and the reduction of 3-fold klt ﬂips to 3-fold pl ﬂips. This material is included here primarily for pedagogical reasons. The section ends with a rather sketchy outline of the construction of 3-fold pl ﬂips.
59 The requirement gets stronger as i → ∞. In practice, I only use the following consequence of uniform asymptotic saturation. If D• is bounded, D = lim Di = sup Di , and saturation holds uniformly on Y , then Mob jDY + CY ≤ jDj Y ≤ jDY for all j. 60 A Shokurov algebra is a bounded canonically a-saturated pbd-algebra. 61 Let (X , B) be a klt pair, and f : X → Z a birational weak Fano contraction to an afﬁne variety Z. Recall what this means: f∗ OX = OZ and −(K + B) is nef and big over Z. All Shokurov algebras on X are ﬁnitely generated.
By deﬁnition, (X , S + B) is plt if and only if A (X , S + B)Y ≥ 0 on all models Y → X . Therefore, if D is effective, D + A (X , S + B)Y is effective on every model. 26 A b-divisor D on X is exceptionally saturated over X if it is E-saturated for all E effective and exceptional over X . 27 The Q-Cartier closure D of a Q-Cartier integral Weil divisor D is exceptionally saturated. Proof The statement is equivalent to the well-known elementary fact that, for all models f : Y → X over X , ai Ei = OX (D) f∗ OY f ∗ (D) + if all Ei are exceptional and all ai ≥ 0.