# Algebraic Curves and Riemann Surfaces by Rick Miranda By Rick Miranda

During this booklet, Miranda takes the method that algebraic curves are top encountered for the 1st time over the advanced numbers, the place the reader's classical instinct approximately surfaces, integration, and different techniques should be introduced into play. accordingly, many examples of algebraic curves are offered within the first chapters. during this method, the booklet starts off as a primer on Riemann surfaces, with advanced charts and meromorphic capabilities 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 provided in an algebraic demeanour, through an edition of the adelic evidence, expressed thoroughly by way of fixing a Mittag-Leffler challenge. Sheaves and cohomology are brought as a unifying gadget within the latter chapters, in order that their application and naturalness are instantly noticeable. Requiring a historical past of a one semester of complicated variable! conception and a 12 months of summary algebra, this can be an outstanding graduate textbook for a second-semester path in complicated variables or a year-long path in algebraic geometry.

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Note that the projection n: X +BY^>X vBY is closed whenever the sections s and t are closed, since if E c X is closed then the saturation n~ 1nE = E + ts~ *£ is closed and similarly if F c Y is closed. Moreover the fibres arefinite,in this case, and so the projection n is proper. The sections s and t also define a triad X->X xBY+- y, where the components of u are (idx, tp) and the components of v are (sq, idy). Since u, v are embeddings so is the fibrewise push-out X vBY^X xBY; Downloaded from University Publishing Online.

Fibrewise pointed topological spaces 43 We regard B as a fibrewise pointed topological space over itself with the identity as section and projection. Moreover we regard B x T as a fibrewise pointed topological space over B, for each pointed topological space T, with section given by b -• (b, t0), where t0 denotes the basepoint. Fibrewise pointed topological spaces over a given base form a category using the continuous fibrewise pointed functions as morphisms. The equivalences in the category are called fibrewise pointed topological equivalences.

Hence the result is obtained. 27). Let : X -+ Y be a proper fibrewise surjection, where X and Y are fibrewise topological over B. If X is fibrewise regular then so is Y. For let X be fibrewise regular. Let y be a point of Yb (b e B) and let V be a neighbourhood of y in Y. Then 0 " * V is a neighbourhood of the compact ^"HjO in X. 24), therefore, there exists a neighbourhood W of b in B and a neighbourhood U of "x (y) in AV such that the closure Xwn U of U in ^V is contained in 0 ~ 1 K Now since 0 ^ is closed there exists a neighbourhood Kr of y in IV s u c h that <\>~lV c= (7, and then the closure AT^n P r of V in AV is contained in V since Cl K' = C l O ^ " 1 * " ) = C l ^ " 1 ^ ' ) cz (j) Cl 1/ cz (jxfi-iVcz V.