By Rick Miranda
During this booklet, Miranda takes the procedure that algebraic curves are most sensible encountered for the 1st time over the advanced numbers, the place the reader's classical instinct approximately surfaces, integration, and different thoughts should be introduced into play. for this reason, many examples of algebraic curves are awarded within the first chapters. during this manner, the ebook starts off as a primer on Riemann surfaces, with advanced charts and meromorphic features taking heart degree. however the major examples come from projective curves, and slowly yet absolutely the textual content strikes towards the algebraic class. Proofs of the Riemann-Roch and Serre Duality Theorems are offered in an algebraic demeanour, through an edition of the adelic facts, expressed thoroughly by way of fixing a Mittag-Leffler challenge. Sheaves and cohomology are brought as a unifying equipment within the latter chapters, in order that their software and naturalness are instantly noticeable. Requiring a heritage of a one semester of advanced variable! thought and a 12 months of summary algebra, this is often an outstanding graduate textbook for a second-semester direction in advanced variables or a year-long path in algebraic geometry.
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Additional resources for Algebraic Curves and Riemann Surfaces
We identify "o and "i with their algebraic counterparts (partial results about the higher homotopy groups are to be found in later chapters). In particular, our study of the fundamental group requires a review of descent techniques in the context of principal G-fibrations. )et with abelian local coefficients is isomorphic to the cohomology of X. with coefficients in the corresponding locally constant sheaf. 8, the topological type itself determines in some sense the cohomology groups of X. with values in any abelian sheaf.
A map of bisimplicial schemes ¢: U .. -+ is V.. over f sucli that x ¢s,t: us,t .... s)t -+ Ys (coskt-l Vs)t for each s,t of X. ), ~ 0. The category of rigid hypercoverings is a left directed category. Proof. : Us. -:>Vs. has the property that the (t-1 )-truncation of ¢s. determines coskt-l ¢s. 1 that there is at most one map between any two rigid hypercoverings over a given map of simplicial schemes. ) is left directed, it thus suffices to observe that if U.. -+ X. and V ...... X. R are rigid hypercoverings then their rigid product U..
Us,t), where the sum is indexed by the set of maps over XS. Consequently, the restriction to us,t of an injective on Xs is injective. Therefore, H*(U.. U .. U .. 5. U .. )) = H*(I(U .. )) = Is(Xs) for each s ~ 0. Consequently, H*(I(U .. )) which equals H*(X. 4. • We now prove that sheaf cohomology can be computed using hypercoverings. -L. Verdier in , V. 1. 8. Let X. be a simplicial scheme. , ) ~ colim H*( (U •• )) where the colimit is indexed by U.. ). Proof. ) and any V .. ), H 0(X. , F)=Ho(F(V ..
Algebraic Curves and Riemann Surfaces by Rick Miranda