By A.N. Parshin, I.R. Shafarevich, Yu.G. Prokhorov, S. Tregub, V.A. Iskovskikh

The purpose of this survey, written by means of V.A. Iskovskikh and Yu.G. Prokhorov, is to supply an exposition of the constitution idea of Fano types, i.e. algebraic vareties with an abundant anticanonical divisor. Such forms clearly look within the birational class of sorts of damaging Kodaira measurement, and they're very with regards to rational ones. This EMS quantity covers various methods to the category of Fano kinds corresponding to the classical Fano-Iskovskikh "double projection" strategy and its variations, the vector bundles approach because of S. Mukai, and the strategy of extremal rays. The authors speak about uniruledness and rational connectedness in addition to contemporary growth in rationality difficulties of Fano types. The appendix includes tables of a few sessions of Fano types. This publication should be very worthy as a reference and examine consultant for researchers and graduate scholars in algebraic geometry.

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22 In this chapter, we only use saturation with D effective and D + C effective. 23 Let X be a normal variety. We say that a property P holds on high models if P holds on a particular model f : Y → X and on every ‘higher’ model, that is, every model Y → Y → X . 24 A b-divisor D on X is C-saturated if DY is CY -saturated on high models Y → X of X . If Y → X is a model of X and DY is CY -saturated, we say that saturation holds on Y . When C = A(X , B) is the discrepancy b-divisor of a klt pair (X , B), we say that D is canonically saturated.

By deﬁnition, the ith graded piece of R is Ri = H 0 X , OX (iDi ) . Note that, by deﬁnition of the sheaf OX (iDi ), Ri is equipped with a natural inclusion Ri ⊂ k(X ), and the convexity of D• ensures that Ri Rj ⊂ Ri+j ; the product in the algebra is inherited from the product in k(X ). Thus, a pbd-algebra is a function algebra. The sequence D• is called the characteristic sequence of the pbd-algebra. We say that the algebra is bounded if it is bounded as a function algebra. This is equivalent to saying that the characteristic sequence is bounded.

2) The canonical divisor of a normal variety X is a b-divisor. Indeed, the divisor K = divX ω of a rational differential ω ∈ k(X ) naturally makes sense as a bdivisor, for Zariski teaches how to take the order of vanishing of ω along a geometric valuation of X . I follow established usage and say that K is the canonical divisor of X when, more precisely, I can only make sense of the canonical divisor class; K is a divisor in the canonical class. (3) If X is a normal variety, and B = bi Bi ⊂ X is a Q-divisor, the discrepancy of the pair (X , B) is the b-divisor A = A(X , B) with trace AY deﬁned by the formula: KY = f ∗ (KX + B) + AY on models f : Y → X of X .