Titlebook: Algebraic Curves; Towards Moduli Space Maxim E. Kazaryan,Sergei K. Lando,Victor V.‘Prasol Textbook 2018 Springer Nature Switzerland AG 2018

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Curves in Projective Spaces,nsional space there is much more freedom. However, to define curves in . and higher dimensional projective spaces is more difficult than in the plane. In this chapter, we discuss methods of defining such curves.

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Differential 1-Forms on Curves,ean primarily spaces of meromorphic functions, vector fields, and differential forms. These spaces are endowed with natural algebraic structures, which allows one to express properties of curves in algebraic terms.

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Exam Problems,hematics of the Higher School of Economics in 2010–2014. Most of these problems were given as exercises in the main text, and we have collected them here for the reader’s convenience. Along with problems, we also give a list of exam questions.

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https://doi.org/10.1007/978-3-030-98132-7Algebraic curves are curves given by polynomial equations in projective spaces. On the other hand, algebraic curves are one-dimensional complex manifolds, and to define them, there is no need to embed them anywhere. We will consider various ways to define curves and discuss how one can decide whether they result in the same curve.

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https://doi.org/10.1007/978-3-642-25544-1The Riemann–Roch theorem establishes a relationship between two numbers: the dimension .(.) of the vector space .(.) of meromorphic functions with divisor ≥−. and the dimension .(.) of the space ..(.) of meromorphic 1-forms with divisor ≥ ..

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Decision Making in Complex Systems,In the first section of this chapter, we give a proof of the Riemann–Roch formula .(.) − .(. − .) = . − . + 1. In the second section, we present a geometric interpretation of the quantities occurring in the Riemann–Roch formula in terms of canonical curves.

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