By Reid M.

This can be a first graduate direction in algebraic geometry. It goals to provide the coed a boost up into the topic on the examine point, with plenty of fascinating issues taken from the type 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 throughout the topic, possibly in parallel with a standard textbook. The early chapters introduce themes which are priceless all through projective and algebraic geometry, make little calls for, and bring about 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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Extra info for Chapters on algebraic surfaces

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If you have trouble with this question, refer to [H1], Chap. II or one of the books on sheaf theory. 28. If F ⊂ G is a subsheaf, construct the quotient sheaf G/F as the associated sheaf of the presheaf U → Γ(U, G)/Γ(U, F), and prove that it has the universal mapping property for maps from G to a sheaf killing F. Prove also that its stalks are GP /FP , so that the sequence 0 → F → G → G/F → 0 is exact. 29. If f : X → Y is a continuous map of topological spaces and F is a sheaf on Y , construct the sheaf theoretic pullback f −1 F, whose stalk at P ∈ X is Ff (P ) .

As a rule, traditional topologists have only allowed maps of constant rank between vector bundles, which is equivalent to saying that the kernel, image and cokernel are locally direct summands. As we have seen in Examples 1–3, the more general notion of sheaf homomorphism between locally free sheaves is very useful in algebraic geometry. 9. Rules of coherent cohomology This table of rules states the main useful results of coherent cohomology at a fairly simple level of generality. I will take them as axioms throughout.

See also, for example, [R1], App. to §2. Exercises to Chapter B 47 viii. Euler–Poincar´ e characteristic χ(X, F) and Hilbert polynomial Whenever the dimensions are finite, I write hi (X, F) = dimk H i (X, F). Define the ∞ Euler–Poincar´e characteristic of F by χ(X, F) = i=0 (−1)i hi (X, F). Although i its definition involves all the cohomology groups H (X, F), this alternating sum is in fact a much more elementary quantity. For example, from the cohomology long exact sequence (iv), it follows at once that χ(X, F) = χ(X, F ′ ) + χ(X, F ′′ ).