By Parshin, Shafarevich

The purpose of this survey, written through V.A. Iskovskikh and Yu.G. Prokhorov, is to supply an exposition of the constitution thought of Fano types, i.e. algebraic vareties with an considerable anticanonical divisor. Such kinds clearly seem within the birational category of sorts of unfavourable Kodaira size, and they're very on the subject of rational ones. This EMS quantity covers various ways to the class of Fano forms corresponding to the classical Fano-Iskovskikh ''double projection'' procedure and its adjustments, the vector bundles process as a result of S. Mukai, and the strategy of extremal rays. The authors talk about uniruledness and rational connectedness in addition to contemporary growth in rationality difficulties of Fano kinds. The appendix includes tables of a few sessions of Fano types. This booklet may be very important as a reference and study consultant for researchers and graduate scholars in algebraic geometry.

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**Example text**

For the proofs, see [FAC]. Anyone complaining that the paper is in French will receive a blast of unpleasant sarcasm. Actually, the hard thing is not to get used to these rules, but to understand what a coherent sheaf is. Data 1 For any variety X over k and any (quasi-) coherent sheaf F on X there is a k-vector space H i (X, F), that is functorial in F. In other words a homomorphism of sheaves of OX -modules a : F → G gives rise to a linear map a∗ : H i (X, F) → H i (X, G), with obvious compatibilities.

1 and Ex. 12. 7. A nonsingular plane curve A of degree a has genus a−1 2 . If A ⊂ Xd is as in Ex. 3, check your answers to Ex. 3 and Ex. 5 against the genus formula. 8. 6. Show that KX = kL and use the genus formula to calculate the selfintersection of any section of X → P1 . 6. 9. Consider affine coordinates x1 , . . 13, s = dx1 ∧ · · · ∧ dxn has a pole of order n + 1 along the hypersurface at infinity x0 . [Hint: Use coordinates y0 , y2 , . . ] 10. 13. 11. Let F(a1 , . . , an ) be the scroll.

Intersection numbers 2. Let {Γi }ki=1 be a bunch of k curves such that Σ = Γi is connected, and suppose that there is a surjective morphism f : X → C to a nonsingular curve C which contracts Σ to a point Q ∈ C. Then every (n1 , . . , nk ) ∈ Zk \ 0 satisfies q(n1 , . . , nk ) = n i Γi 2 ≤ 0. In other words, q is negative semidefinite. Moreover, q(n1 , . . , nk ) = 0 holds if and only if ni Γi is proportional to the fibre. More precisely, if t ∈ mQ ⊂ OC,Q is a local parameter at Q, and g = f ∗ (t) ∈ k(X) is the rational function on X obtained as the pullback of t, then ni Γi and the connected component of div t at f −1 Q are rational multiples of one another.