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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,Proof of the Riemann–Roch Formula,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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Stable Curves,In the previous chapter, we introduced the notion of a stable rational curve with marked points. The (modular) stability of a curve means that it has a finite group of automorphisms.

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https://doi.org/10.1007/978-3-030-02943-2algebraic curves; Riemann-Roch theorem; Weierstrass points; Abel theorem; moduli spaces; compactification

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Decision Making for Energy Futuresann surface is a two-dimensional oriented surface; its topological properties are uniquely determined by a nonnegative integer, the genus. At the same time, individual characteristics of algebraic curves are complicated, and two different curves, even of the same genus, usually bear little resemblan

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Framing Health Security Decisions,urface, the topology is uniquely determined by its genus (or, equivalently, its Euler characteristic). However, along with a topological structure, a curve has a complex structure. It singles out analytic functions among all the functions on the curve.

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Framing Health Security Decisions,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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Waymond Rodgers,Timothy G McFarlinnto other complex curves, first of all, one-to-one mappings from a complex curve to itself, i.e., automorphisms of a curve. All automorphisms of a given curve form a group. For a curve of genus 0 (projective line), this group is three-dimensional. For any curve of genus 1 (elliptic curve), it is one