By V. I. Danilov (auth.), I. R. Shafarevich (eds.)
This EMS quantity contains elements. the 1st half is dedicated to the exposition of the cohomology conception of algebraic kinds. the second one half offers with algebraic surfaces. The authors, who're recognized specialists within the box, have taken pains to offer the fabric conscientiously and coherently. The e-book comprises a variety of examples and insights on a number of themes. This e-book can be immensely beneficial to mathematicians and graduate scholars operating in algebraic geometry, mathematics algebraic geometry, advanced research and similar fields.
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Additional info for Algebraic Geometry II: Cohomology of Algebraic Varieties. Algebraic Surfaces
We will give an application of the Riemann-Roch theorem and the duality to projective surfaces. Theorem. Let H be an arbitrary hyperplane sections of a surface X, and D a divisor on X such that the intersection number (D. H) equals 0. Then (D. D) ::; 0, and if (D. D) = 0 then Dis numerically equivalent to zero (i. , (D. C) = 0 for every C c X). We will give another equivalent statement of the theorem, which also explains its name. Let N(X) denote the quotient of the divisor group Div X by the subgroup of divisors numerically equivalent to zero.
Theorem. Let F be a quasi-coherent sheaf on an affine va'riety X. Then Hq(X, F) = 0 for all q > 0. One can show that converse is also true in the Noetherian case: if the cohomology of every quasi-coherent sheaf on a scheme X are trivial, then X is an affine scheme (Hartshorne (1977)). In particular, a scheme is affine if and only if its components are affine. Since Serre's theorem plays an important role, we will sketch its proof. Let A denote a dass of open subsets of the form D(f) of an affine scheme X = Spec A, where f E A.
Theorem (Grothendieek). For every a E K(X) eh(fka) ·td(Ty) = f*(eh(a) ·td(Tx)). 6. Principle of the Proof. As it often happens, it is easier to prove a general formula than its special case - the formula helps itself. In the case in question, we are dealing with a morphism instead of a fixed variety. Clearly, if Grothendieek's formula is valid for morphisms f: X ---+ Y and g: Y ---+ Z, then it also valid for the eomposition g o f: X ---+ Z. This follows essentially from the Leray spectral sequenee, which gives (g o f)k = 9k o fk.