Algebraic curves. Towards moduli spaces. Translated from the Russian by Natalia Tsilevich
This book focuses on the theory of Riemann surfaces from the modern viewpoint of complex algebraic curves and their moduli spaces. Chapters 1--3 introduce preliminaries of curves, including their algebraic structure, complex structure, topology, branched coverings, realization in the plane, and rational parametrization. Chapters 4--6 study curves in projective spaces, Plücker Formulas, and mappings of curves with a focus on elliptic curves. Chapters 7--12 highlight classical objects associated with curves, including differential forms, line bundles and divisors, Riemann-Roch Formula, Weierstrass points, and Jacobian. Chapters 13--18 focus on various moduli spaces, including Hilbert schemes, moduli spaces of curves with marked points, and moduli spaces of stable maps. Finally in Chapter 19 many exam questions are given to test the reader's understanding of the subject.
scientific article; zbMATH DE number 6982465