Notes (By Sections of "The Rising Sea" by Vakil)
Chapter 3: Towards Affine Schemes
The main algebro-geometric objects we will be studying in this text are \tcrb{schemes}. We'll motivate their construction along the way as they have an especially peculiar presentation in comparison with other typical definitions of geometric objects in mathematics. The local models of schemes are \tcrb{affine schemes} and we will be introducing what these spaces are, study their interesting topological properties, and discover their significance. We will be "unveiling" the geometry behind algebra.
A quote by C. McLarty included in Vakil's notes implies that historically the definition of schemes was not really the monumental invention of Grothendieck himself. Rather many experts at the time knew such a structure could be readily accessed. The real innovation was \textit{how} and \textit{why} Grothendieck thought it so deeply necessary to develop this language in the 1000 pages of the EGA to clarify the nature of the results of previous algebraic geometers. This is to say that the meat and potatoes of the theory of schemes is not in the form of its definition. It is the perspectives one now is allowed to access after the fact. Giving rigorous and solid foundations for very deep analysis of what causes certain theorems to take form.
I myself am excited, after about 200 or so pages of text defining and playing around with the "not-so-innovative" construction of schemes, to get to the meat and potatoes.
Notes (by sections from "Undergraduate Algebraic Geometry" by Miles Reid)
Chapter 1.1-1.6: Parametrizations, Rational Points, and Classification of Conics
In this section we start by doing some examples of simple parametrizations of curves in R^2. In particular, curves which are cut out by degree two polynomial equations. One of the natural questions posed in number theory is: When given an expression, what are all the integral solutions? In particular, the first problem we explore is using parametrizations of the unit circle to find all integral Pythagorean triples. Via very simple calculations this problem can be solved very nicely using geometric methods. This method can then generalize to many other similar problems like integral points of ellipses. With this example to get us used to working in charts, we then turn to the slightly more complicated problem of classifying all spaces in R^3 which can be cut out by a quadratic polynomial in two variables. Using techniques of homogenization and translating the problem into the projective context we see that the tools of modern algebra, specifically the correspondence of symmetric bilinear forms and homogeneous quadratics in three variables, we find that the seemingly mysterious classification of conic sections from high school math is nothing but a classification by signatures of symmetric 3x3 matrices! This kicks off our exciting study of algebraic geometry with the solutions of two very different flavors of problems: 1) Number theoretic existence problems and 2) Classification of geometric objects using projective geometry.