Topics covered: Algebraic Sets, The Hilbert basis theorem, The Zariski topology, The Hilbert Nullstellensatz, Regular functions, Noether normalization theorem, Dimension,  Affine Algebraic Varieties, Sheaves,  Ringed spaces, , Morphisms of ringed spaces, Affine algebraic varieties, Subvarieties, Birational equivalence,  Noether Normalization Theorem, Local Study, Tangent spaces to plane curves, Tangent cones to plane curves, The local ring at a point on a curve, The differential of a regular map, Tangent spaces to affine algebraic varieties,   Algebraic varieties,  Products of varieties,   Rational maps; birational  equivalence,   Local study,  Smooth maps, Projective Varieties Algebraic subsets of , The homogeneous coordinate ring of a projective variety, Regular functions on a  projective variety, Maps from projective varieties,  Bezout’s theorem.