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These lectures, delivered by Professor Mumford at Harvard in 1963-1964, are devoted to a study of properties of families of algebraic curves, on a non-singular projective algebraic curve defined over an algebraically closed field of arbitrary characteristic. The methods and techniques of Grothendieck, which have so changed the character of algebraic geometry in recent years, are used systematically throughout. Thus the classical material is presented from a new viewpoint.

Curves, Algebraic. --- Surfaces, Algebraic. --- Affine space. --- Algebraic geometry. --- Algebraic topology. --- Algebraically closed field. --- Binary operation. --- Chern class. --- Coherent sheaf. --- Cohomology. --- Complex vector bundle. --- Dense set. --- Differential form. --- Direct product. --- Family of curves. --- Formal power series. --- Functor. --- Generic point. --- Group ring. --- Homomorphism. --- Invertible sheaf. --- Local ring. --- Morphism of schemes. --- Morphism. --- Nonlinear system. --- Open set. --- Pairwise. --- Polynomial. --- Power series. --- Projective space. --- Rational function. --- Rational point. --- Sheaf (mathematics). --- Subring. --- Summation. --- Symmetric function. --- Topology. --- Union (set theory). --- Zariski topology. --- Zero divisor.

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This book builds upon the revolutionary discovery made in 1974 that when one passes from function f to a function J of paths joining two points A1≠A1 the connectivities R1 of the domain of f can be replaced by connectivities R1 over Q, common to the pathwise components of a basic Frechet space of classes of equivalent curves joining A1 to A1. The connectivities R1, termed "Frechet numbers," are proved independent of the choice of A1 ≠ A1, and of a replacement of Mn by any differential manifold homeomorphic to Mn.Originally published in 1976.The Princeton Legacy Library uses the latest print-on-demand technology to again make available previously out-of-print books from the distinguished backlist of Princeton University Press. These editions preserve the original texts of these important books while presenting them in durable paperback and hardcover editions. The goal of the Princeton Legacy Library is to vastly increase access to the rich scholarly heritage found in the thousands of books published by Princeton University Press since its founding in 1905.

Differentiable manifolds. --- Global analysis (Mathematics) --- Calculus of variations. --- Algebraic topology. --- Analytic function. --- Arc length. --- Axiom. --- Bernhard Riemann. --- Boundary value problem. --- Cartesian coordinate system. --- Coefficient. --- Compact space. --- Computation. --- Conjugate points. --- Connectivity (graph theory). --- Continuous function. --- Corollary. --- Countable set. --- Counting. --- Cramer's rule. --- Curve. --- Deformation theory. --- Degeneracy (mathematics). --- Derivative. --- Diffeomorphism. --- Differentiable manifold. --- Differential equation. --- Differential geometry. --- Differential structure. --- Dimension. --- Domain of a function. --- Eilenberg. --- Einstein notation. --- Equation. --- Euclidean space. --- Euler characteristic. --- Euler equations (fluid dynamics). --- Euler integral. --- Existence theorem. --- Existential quantification. --- Exotic sphere. --- Family of curves. --- Finite set. --- First variation. --- Geometry. --- Global analysis. --- Homeomorphism. --- Homology (mathematics). --- Homotopy. --- Implicit function theorem. --- Inference. --- Integer. --- Intersection (set theory). --- Interval (mathematics). --- Invertible matrix. --- Jacobian matrix and determinant. --- Lagrange multiplier. --- Linear combination. --- Linear map. --- Line–line intersection. --- Mathematical proof. --- Maximal set. --- Metric space. --- N-sphere. --- Neighbourhood (mathematics). --- Null vector. --- Open set. --- Pairwise. --- Parameter. --- Parametric equation. --- Parametrization. --- Partial derivative. --- Partial function. --- Phase space. --- Positive definiteness. --- Projective plane. --- Quadratic form. --- Quadratic. --- Rate of convergence. --- Rational number. --- Real variable. --- Resultant. --- Riemannian manifold. --- Scientific notation. --- Sign (mathematics). --- Special case. --- Sturm separation theorem. --- Submanifold. --- Subsequence. --- Subset. --- Taylor's theorem. --- Tensor algebra. --- Theorem. --- Theory. --- Topological manifold. --- Topological space. --- Topology. --- Tuple. --- Unit vector. --- Variable (mathematics). --- Variational analysis. --- Weierstrass function. --- Without loss of generality.