By Reid M.

This can be a first graduate direction in algebraic geometry. It goals to provide the scholar a boost up into the topic on the learn point, with plenty of fascinating themes taken from the category of surfaces, and a human-oriented dialogue of a few of the technical foundations, yet without pretence at an exhaustive remedy. i'm hoping that graduate scholars can use a few of these chapters as a reader in the course of the topic, perhaps in parallel with a traditional textbook. The early chapters introduce themes which are important all through projective and algebraic geometry, make little calls for, and result in enjoyable calculations. The intermediate chapters introduce parts of the technical language steadily, while the later chapters get into the substance of the type of surfaces.

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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.