This book introduces the contemporary notions of algebraic varieties, morphisms of varieties, and adeles to the classical subject of plane curves over algebraically closed fields. It is useful for advanced undergraduate and beginning graduate students in mathematics.
Preface
PREREQUISITES
SOME FACTS ABOUT POLYNOMIALS
AFFINE PLANE CURVES
TANGENT SPACES
THE LOCAL RING AT THE POINT
PROJECTIVE PLANE CURVES
RATIONAL MAPPINGS, BIRATIONAL CORRESPPONDENCES AND ISOMOPHISMS OF CURVES
EXAMPLES OF RATIONAL CURVES
THE CORRESPONDENCE BETWEEN VALUATIONS AND POINTS
AN OVERVIEW AND SIDEWAYS GLANCE
DIVISORS
THE DIVISOR OF A FUNCTION HAS DEGREE 0
RIEMANN’S THEOREM
THE GENUS OF A NONSINGULAR PLANE CURVE
CURVES OF GENUS 0 AND 1
A CLASSIFICATION OF ISOMORPHISM CLASSES OF CURVES OF GENUS 1
THE GENUS OF A SINGULAR CURVE
INFLECTION POINTS ON PLANE CURVES
BEZOUT’S THEOREM
ADDITION ON A NONSINGULAR CUBIC
DERIVATIONS, DIFFERENTIALS AND THE CANONICAL CLASS
ADELES AND THE RIEMANN-ROCH THEROEM
Bibliography
Notation
Index
Biography
C. Orzech
