I: Examples of Varieties and Maps. - Lecture 1 Affine and Projective Varieties. - Lecture 2 Regular Functions and Maps. - Lecture 3 Cones Projections and More About Products. - Lecture 4 Families and Parameter Spaces. - Lecture 5 Ideals of Varieties Irreducible Decomposition and the Nullstellensatz. - Lecture 6 Grassmannians and Related Varieties. - Lecture 7 Rational Functions and Rational Maps. - Lecture 8 More Examples. - Lecture 9 Determinantal Varieties. - Lecture 10 Algebraic Groups. - II: Attributes of Varieties. - Lecture 11 Definitions of Dimension and Elementary Examples. - Lecture 12 More Dimension Computations. - Lecture 13 Hilbert Polynomials. - Lecture 14 Smoothness and Tangent Spaces. - Lecture 15 Gauss Maps Tangential and Dual Varieties. - Lecture 16 Tangent Spaces to Grassmannians. - Lecture 17 Further Topics Involving Smoothness and Tangent Spaces. - Lecture 18 Degree. - Lecture 19 Further Examples and Applications of Degree. - Lecture 20 Singular Points and Tangent Cones. - Lecture 21 Parameter Spaces and Moduli Spaces. - Lecture 22 Quadrics. - Hints for Selected Exercises. - References. Language: English