Lecture Notes Algebraic Geometry III/IV by Matt Kerr

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Lecture Notes Algebraic Geometry III/IV written by Matt Kerr
. This is an other book of mathematics cover the following topics.

Introduction and Motivation Two theorems on conics in the plane, Riemann surfaces and algebraic curves, The normalization theorem, Lines, conics, and duality

General definitions and results Complex manifolds and algebraic varieties, More on projective algebraic varieties, Smooth varieties as complex manifolds, The connectedness of algebraic curves, Hilbert’s nullstellensatz, Local analytic factorization of polynomials, Proof of the normalization theorem, Intersections of curves, Meromorphic 1-forms on a Riemann surface, The genus formula, Some applications of Bézout

Cubic curves The singular cubic, Putting a nonsingular cubic in standard form, Canonical normalization of the Weierstrass cubic, Group law on the nonsingular cubic, Abel’s theorem for elliptic curves, The Poncelet problem, Periods of families of elliptic curves, Counting Fp-rational points on elliptic curves

Curves of higher genus The algebraicity of global analytic objects, The Riemann-Roch Theorem, Applications of Riemann-Roch, I: special Riemann surfaces, Applications of Riemann-Roch, II: general Riemann surfaces, Abel’s Theorem, part I, Abel’s Theorem, part II