Algebraic Geometry

Introductory Geometry: Algebraic Geometry

Algebraic Geometry

Physical objects and constraints may be modeled by polynomial equations and inequalities. For this reason algebraic geometry, the study of solutions to systems of polynomial equations, is a tool for scientists and engineers. Moreover, relations between concepts arising in science and engineering are often described by polynomials. Whatever their source, once polynomials enter the picture, notio...

Algebraic Geometry

Algebraic Geometry

Examples • If S = ∅, then Z(S) = An. • If S = {1}, then Z(S) = ∅. • If S = {x1 − a1, . . . , xn − an}, then Z(S) = {(a1, . . . , an)}. Remark. All rings in this course are commutative and have 1. Remark. If A is a ring, then any subset S ⊆ A generates a minimal ideal 〈S〉 ⊆ A. In fact, we have 〈S〉 = {∑ aj xj : aj ∈ A, xj ∈ S}. Lemma. Z(S) = Z(〈S〉) for all S ⊆ k[x1 , . . . , xn]. Proof. Since 〈S〉...

Algebraic Geometry

These notes are an introduction to the theory of algebraic varieties. In contrast to most such accounts they study abstract algebraic varieties, and not just subvarieties of affine and projective space. This approach leads more naturally into scheme theory. v2.01 (August 24, 1996). First version on the web. v3.01 (June 13, 1998). v4.00 (October 30, 2003). Fixed errors; many minor revisions; add...