Local invariants of an algebraic variety in a point. The tangent space. Singular and nonsingular points of an algebraic variety. Topological properties of projective algebraic varieties and applications. The Segre embedding and products of algebraic varieties. Applications. The dimension theory of algebraic varieties and applications. Power series and the Weierstrass Preparation Theorem. Applications. The factoriality of the local ring of an algebraic variety and applications. Divisors and differential forms on a nonsingular algebraic variety. The Riemann-Roch theorem for nonsingular projective curves.