This book, based on lectures presented in courses on algebraic geometry taught by the author at Purdue University, is intended for engineers and scientists (especially computer scientists), as well as graduate students and advanced undergraduates in mathematics. In addition to providing a concrete or algorithmic approach to algebraic geometry, the author also attempts to motivate and explain its link to more modern algebraic geometry based on abstract algebra.The book covers various topics in the theory of algebraic curves and surfaces, such as rational and polynomial parametrization, functions and differentials on a curve, branches and valuations, and resolution of singularities. The emphasis is on presenting heuristic ideas and suggestive arguments rather than formal proofs. Readers will gain new insight into the subject of algebraic geometry in a way that should increase appreciation of modern treatments of the subject, as well as enhance its utility in applications in science and industry.
Rational and polynomial parametrizations Fractional linear transformations Cubic curves Cubic surfaces and general hypersurfaces Outline of the theory of plane curves Affine plane and projective plane Sphere with handles Functions and differentials on a curve Polynomials and power series Review of abstract algebra Some commutative algebra Hensel's lemma and Newton's theorem More about Newton's theorem Branches and valuations Divisors of functions and differentials Weierstrass preparation theorem Intersection multiplicity Resolution of singularities of plane curves Infinitely near singularities Parametrizing a quartic with three double points Characteristic pairs Criterion for one place and Jacobian problem Inversion formula and Jacobian problem Surfaces Hypersurfaces Resolution of singularities of algebraic surfaces Birational and polyrational transformations Valuations and birational correspondence Rational cylinders through a variety Resultants.