Homogeneous and non-homogeneous point coordinates -- Coordinate rings of irreducible varieties -- Normal varieties -- Divisorial cycles on a normal projective variety V/k (dim(V)=r?1) -- Linear systems -- Divisors on an arbitrary variety V -- Intersection theory on algebraic surfaces (k algebraically closed) -- Differentials -- The canonical system on a variety V -- Trace of a differential -- The arithemetic genus -- Normalization and complete systems -- The Hilbert characteristic function and the arithmetic genus of a variety -- The Riemann-Roch theorem -- Subadjoint polynomials -- Proof of the fundamental lemma.