Affine and projective algebraic sets. Non-singular points and tangent space. Dimension and degree of a projective variety. Hypersurfaces in projective space. Linear systems of hypersurfaces. The Jacobian group of a linear system on a line. Algebraic correspondences between lines. Birational plane model of a projective curve. Plane curves. Cusps, flexes and ordinary multiple points. Bézout’s theorem. The genus of a plane curve (according to Riemann). Rational curves. Rational normal curves. Elliptic normal curves. Plane cubic curves. Modulus of a plane cubic. Structure of abelian group on such curves. Modulus of a plane cubic. Classification of degree 3 projective curves.