Introductory Geometry: Algebraic Geometry

Introductory GeometryCourse No. 100 351 Fall 2005 Second Part: Algebraic GeometryMichael StollContents1. What Is Algebraic Geometry ?22. Affine Spaces and Algebraic Sets33. Projective Spaces and Algebraic Sets64. Projective Closure and Affine Patches95. Morphisms and Rational Maps116. Curves Local Properties147. B ezout s Is Algebraic Geometry ?Linear Algebracan be seen (in parts at least) as the study of systems of linearequations. In geometric terms, this can be interpreted as the study of linear (oraffine) subspaces ofCn(say). Algebraic Geometrygeneralizes this in a natural way be looking at systems ofpolynomialequations. Their geometric realizations (their solution sets inCn, say)are calledalgebraic questions one can study in various parts of mathematics lead in a naturalway to (systems of) polynomial equations, to which the methods of AlgebraicGeometry can be Geometry provides a translation betweenalgebra(solutions of equations)andgeometry(points on Algebraic varieties).

2 1. What Is Algebraic Geometry? Linear Algebra can be seen (in parts at least) as the study of systems of linear equations. In geometric terms, this can be interpreted as the study of linear (or

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