LEADER 04835nam 22006015 450 001 9910309661403321 005 20200703121701.0 010 $a3-030-02943-3 024 7 $a10.1007/978-3-030-02943-2 035 $a(CKB)4100000007522415 035 $a(DE-He213)978-3-030-02943-2 035 $a(MiAaPQ)EBC6311737 035 $a(PPN)23379915X 035 $a(EXLCZ)994100000007522415 100 $a20190121d2018 u| 0 101 0 $aeng 135 $aurnn#008mamaa 181 $ctxt$2rdacontent 182 $cc$2rdamedia 183 $acr$2rdacarrier 200 10$aAlgebraic Curves$b[electronic resource] $eTowards Moduli Spaces /$fby Maxim E. Kazaryan, Sergei K. Lando, Victor V. Prasolov 205 $a1st ed. 2018. 210 1$aCham :$cSpringer International Publishing :$cImprint: Springer,$d2018. 215 $a1 online resource (XIV, 231 p. 37 illus.) 225 1 $aMoscow Lectures,$x2522-0314 ;$v2 311 $a3-030-02942-5 320 $aIncludes bibliographical references and index. 327 $aIntroduction -- 1 Preliminaries ? 2 Algebraic curves -- 3 Complex structure and the topology of curves -- 4 Curves in projective spaces -- 5 Plücker formulas -- 6 Mappings of curves -- 7 Differential 1-forms on curves -- 8 Line bundles, linear systems, and divisors -- 9 Riemann?Roch formula and its applications -- 10 Proof of the Riemann?Roch formula -- 11 Weierstrass points -- 12 Abel?s theorem -- 13 Examples of moduli spaces -- 14 Approaches to constructing moduli spaces -- 15 Moduli spaces of rational curves with marked points -- 16 Stable curves -- 17 A backward look from the viewpoint of characteristic classes -- 18 Moduli spaces of stable maps -- 19 Exam problems -- References -- Index. 330 $aThis book offers a concise yet thorough introduction to the notion of moduli spaces of complex algebraic curves. Over the last few decades, this notion has become central not only in algebraic geometry, but in mathematical physics, including string theory, as well. The book begins by studying individual smooth algebraic curves, including the most beautiful ones, before addressing families of curves. Studying families of algebraic curves often proves to be more efficient than studying individual curves: these families and their total spaces can still be smooth, even if there are singular curves among their members. A major discovery of the 20th century, attributed to P. Deligne and D. Mumford, was that curves with only mild singularities form smooth compact moduli spaces. An unexpected byproduct of this discovery was the realization that the analysis of more complex curve singularities is not a necessary step in understanding the geometry of the moduli spaces. The book does not use the sophisticated machinery of modern algebraic geometry, and most classical objects related to curves ? such as Jacobian, space of holomorphic differentials, the Riemann-Roch theorem, and Weierstrass points ? are treated at a basic level that does not require a profound command of algebraic geometry, but which is sufficient for extending them to vector bundles and other geometric objects associated to moduli spaces. Nevertheless, it offers clear information on the construction of the moduli spaces, and provides readers with tools for practical operations with this notion. Based on several lecture courses given by the authors at the Independent University of Moscow and Higher School of Economics, the book also includes a wealth of problems, making it suitable not only for individual research, but also as a textbook for undergraduate and graduate coursework. 410 0$aMoscow Lectures,$x2522-0314 ;$v2 606 $aAlgebraic geometry 606 $aFunctions of complex variables 606 $aMathematical physics 606 $aAlgebraic Geometry$3https://scigraph.springernature.com/ontologies/product-market-codes/M11019 606 $aFunctions of a Complex Variable$3https://scigraph.springernature.com/ontologies/product-market-codes/M12074 606 $aMathematical Physics$3https://scigraph.springernature.com/ontologies/product-market-codes/M35000 615 0$aAlgebraic geometry. 615 0$aFunctions of complex variables. 615 0$aMathematical physics. 615 14$aAlgebraic Geometry. 615 24$aFunctions of a Complex Variable. 615 24$aMathematical Physics. 676 $a516.352 700 $aKazaryan$b Maxim E$4aut$4http://id.loc.gov/vocabulary/relators/aut$0768199 702 $aLando$b Sergei K$4aut$4http://id.loc.gov/vocabulary/relators/aut 702 $aPrasolov$b Victor V$4aut$4http://id.loc.gov/vocabulary/relators/aut 801 0$bMiAaPQ 801 1$bMiAaPQ 801 2$bMiAaPQ 906 $aBOOK 912 $a9910309661403321 996 $aAlgebraic Curves$92163210 997 $aUNINA