Algebraic geometry is thus often described as the study of those geometric objects that can be described by polynomials. The unit circle centered at the origin (□, □) in the plane satisfying the polynomial □2 + □ 2 − 1 = 0, an algebraic object. For example, the circle, a geometric object, can also be described as the points 0-1:circleįigure 1. Algebraic geometry As the name suggests, algebraic geometry is the linking of algebra to geometry. Invertible Sheaves and Divisors Basic Homology and CohomologyĪppendix A. ![]() Graded Rings and Homogeneous Ideals Projective Varietiesįunctions on Projective Varieties ExamplesĭRAFT COPY: Complied on February 4, 2010. Definition of Projective □-space ℙ□ (□) The Zariski Topology Points and Local rings Tangent SpacesĬhapter 5. Variety as Irreducible: Prime Ideals SubvarietiesĤ.9. Hilbert Basis Theorem Hilbert Nullstellensatz ![]() Zero Sets of PolynomialsĪlgebraic Sets Zero Sets via □ (□) Functions on Zero Sets and the Coordinate Ring The Riemann-Roch Theorem Singularities and Blowing UpĬhapter 4. Higher Degree Curves as Surfaces B´ezout’s Theorem Regular Functions and Function Fields The Group Law for a Smooth Cubic in Canonical Form Cubics as ToriĬross-Ratios and the j-Invariant Cross Ratio: A Projective Invariant The □-InvariantĪlgebraic Geometry: A Problem Solving ApproachĬhapter 3. Projective Change of Coordinates The Complex Projective Line ℙ1Įllipses, Hyperbolas, and Parabolas as Spheres Degenerate Conics - Crossing lines and double lines. Changes of CoordinatesĬonics over the Complex Numbers The Complex Projective Plane ℙ2 Project Lead Tom Garrity Williams CollegeĬhapter 1. The last chapter is on sheaves and cohomology, providing a hint of current work in algebraic geometry.Īlgebraic Geometry A Problem Solving Approach Park City Mathematics Institute 2008 Undergraduate Faculty Program Abstract algebra now plays a critical role, making a first course in abstract algebra necessary from this point on. Chapters 4 and 5 introduce geometric objects of higher dimension than curves. Both chapters are appropriate for people who have taken multivariable calculus and linear algebra. Chapter 2 leads the reader to an understanding of the basics of cubic curves, while Chapter 3 introduces higher degree curves. The first chapter on conics is appropriate for first-year college students (and many high school students). This text consists of a series of exercises, plus some background information and explanations, starting with conics and ending with sheaves and cohomology. It is not an easy field to break into, despite its humble beginnings in the study of circles, ellipses, hyperbolas, and parabolas. ![]() Algebraic Geometry has been at the center of much of mathematics for hundreds of years.
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