Preface Acknowledgments Using This Book PART ��EXAMPLES OF VARIETIES AND MAPS LECTURE 1 Affine and Projective Varieties A Note About Our Field Affine Space andAffineVarieties Projective Space and Projective Varieties Linear Spaces Finite Sets Hypersurfaces Analytic Subvarieties and Submanifolds The Twisted Cubic Rational Normal Curves Determinantai Representation of the Rational Normal Curve Another Parametrization of the Rational Normal Curve The Family of Plane Conics A Synthetic Construction of the Rational Normal Curve 0ther Rational Curves Varieties Defined over Subfields of K A Note on Dimension��Smoothness��and Degree LECTURE 2 Regular Functions and Maps The Zariski Topology Regular Functions on an Affine Variety Projective Varieties Regular Maps The Veronese Map Determinantal ReDresentatiOn of Veronese Varieties Subvarieties of Veronese Varieties The Segre Maps Subvarieties of Segre Varieties Products of Varieties Graphs Fiber Products Combinations of Veronese and Segre Maps LECTURE 3 Cones��Projections��and More About Products Cones Quadrics Projections M0re Cones More Projections Constructible Sets LECTURE 4 Families and Parameter Spaces Families of Varieties The Universal Hyperplane The Universal Hyperplane Section Parameter Spaces of Hypersurfaces Universal Families of Hypersurfaces A Family of Lines ����