By Qing Liu

Advent; 1. a few issues in commutative algebra; 2. common homes of schemes; three. Morphisms and base swap; four. a few neighborhood houses; five. Coherent sheaves and Cech cohmology; 6. Sheaves of differentials; 7. Divisors and purposes to curves; eight. Birational geometry of surfaces; nine. general surfaces; 10. relief of algebraic curves; Bibilography; Index

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A) Let An = ∩m>n πm,n (Am ). Show that (An , πn−1 |An )n is an inverse lim An is bijective. system and that the canonical map ←− lim An → ←− n n (b) Let us suppose that (An , πn )n satisﬁes the Mittag–Leﬄer condition and that An = ∅ for all n. Show that An = ∅, An+1 → An is surjective, and that ←− lim An = ∅. Deduce from this that ←− lim An = ∅. 3. Formal completion 25 ρ → (Cn )n → 0 be an exact sequence (c) Let 0 → (An )n → (Bn )n − of inverse systems of Abelian groups such that (An )n satisﬁes the lim Cn and Xn = ρ−1 Mittag–Leﬄer condition.

A) Let P1 (T1 ) be a generator of m ∩ k[T1 ] and k1 = k[T1 ]/(P1 ). Show that we have an exact sequence 0 → P1 (T1 )k[T1 , . . , Tn ] → k[T1 , . . , Tn ] → k1 [T2 , . . , Tn ] → 0. (b) Show that there exist n polynomials P1 (T1 ), P2 (T1 , T2 ), . . , Pn (T1 , . . , Tn ) such that k[T1 , . . , Ti ] ∩ m = (P1 , . . , Pi ) for all i ≤ n. In particular, m is generated by n elements. 2. 6. We say that a topological space X is quasi-compact if from any open covering {Ui }i of X, we can extract a ﬁnite subcovering.

Tn − αn . Proof By the preceding corollary, k → k[T1 , . . , Tn ]/m is an isomorphism. Let αi ∈ k be the image of Ti in k[T1 , . . , Tn ]/m. Then Ti − αi ∈ m. It follows that m contains the ideal (T1 − α1 , . . , Tn − αn ). Since the latter is maximal, there is equality. The uniqueness of the αi is immediate. 14. Let k be an algebraically closed ﬁeld. Let P1 (T ), . . , Pm (T ) be polynomials in k[T1 , . . , Tn ]. Let Z(P1 , . . , Pm ) := {(α1 , . . , αn ) ∈ k n Pj (α1 , . . , αn ) = 0, 1 ≤ j ≤ m}.