Elementary Algebraic Geometry

I Examples of curves.- 1 Introduction.- 2 The topology of a few specific plane curves.- 3 Intersecting curves.- 4 Curves over ?.- II Plane curves.- 1 Projective spaces.- 2 Affine and projective varieties; examples.- 3 Implicit mapping theorems.- 4 Some local structure of plane curves.- 5 Sphere coverings.- 6 The dimension theorem for plane curves.- 7 A Jacobian criterion for nonsingularity.- 8 Curves in ?2(?) are connected.- 9 Algebraic curves are orientable.- 10 The genus formula for nonsingular curves.- III Commutative ring theory and algebraic geometry.- 1 Introduction.- 2 Some basic lattice-theoretic properties of varieties and ideals.- 3 The Hilbert basis theorem.- 4 Some basic decomposition theorems on ideals and varieties.- 5 The Nullstellensatz: Statement and consequences.- 6 Proof of the Nullstellensatz.- 7 Quotient rings and subvarieties.- 8 Isomorphic coordinate rings and varieties.- 9 Induced lattice properties of coordinate ring surjections; examples.- 10 Induced lattice properties of coordinate ring injections.- 11 Geometry of coordinate ring extensions.- IV Varieties of arbitrary dimension.- 1 Introduction.- 2 Dimension of arbitrary varieties.- 3 The dimension theorem.- 4 A Jacobian criterion for nonsingularity.- 5 Connectedness and orientability.- 6 Multiplicity.- 7 Bezout's theorem.- V Some elementary mathematics on curves.- 1 Introduction.- 2 Valuation rings.- 3 Local rings.- 4 A ring-theoretic characterization of nonsingularity.- 5 Ideal theory on a nonsingular curve.- 6 Some elementary function theory on a nonsingular curve.- 7 The Riemann-Roch theorem.- Notation index.