reference request - Best way to learn Algebraic Geometry?
The trouble with algebraic geometry is that it is, in its modern form, essentially just generalised commutative algebra. Indeed, in a very precise sense, a scheme can be thought of as a generalised local ring. (The structure sheaf OX O X of a scheme X X is a local ring object in the sheaf topos Sh(X) Sh (X), and a OX O X -module is literally a ...
Why study Algebraic Geometry? - Mathematics Stack Exchange
2012年12月10日 · So when any software plots a transcendental surface (or manifold), it is actually displaying a polynomial approximation (an algebraic variety). So the study of algebraic geometry in the applied and computational sense is fundamental for the rest of geometry. From a pure mathematics perspective, the case of projective complex algebraic geometry ...
Newest 'algebraic-geometry' Questions - Mathematics Stack …
Questions tagged [algebraic-geometry] The study of geometric objects defined by polynomial equations, as well as their generalizations: algebraic curves, such as elliptic curves, and more generally algebraic varieties, schemes, etc. Problems under this tag typically involve techniques of abstract algebra or complex-analytic methods. This tag ...
Reference for Algebraic Geometry - Mathematics Stack Exchange
2011年6月22日 · This is an introduction to algebraic geometry through Riemann surfaces/complex algebraic curves which does the kind of thing you want: it proves the equivalence of the different notions of genus, proves Riemann-Roch and explains differentials. Beltrametti et al. - "Lectures on Curves, Surfaces and Projective Varieties".
What is the geometry in algebraic geometry? - Mathematics Stack …
2022年10月26日 · 1. Algebraic geometry is mainly studying algebraic varieties (and their generalizations) per se. So here, algebraic varieties are analogous to Riemannian manifolds and symplectic manifolds in your examples, rather than underlying smooth manifolds. The "geometry" of smooth manifolds is differential topology, roughly speaking, and the "geometry ...
Best Algebraic Geometry text book? (other than Hartshorne)
BEST ON SCHEMES:Görtz; Wedhorn- Algebraic Geometry I, Schemes with Examples and Exercises. Tons of stuff on schemes; more complete than Mumford's Red Book. It does a great job complementing Hartshorne's treatment of schemes, above all because of the more solvable exercises. A second volume is on its way on cohomology.
"Real"-life applications of algebraic geometry
Here's an example of a ``real-life'' application of algebraic geometry. Consider an optimal control problem that adheres to the Karush-Kuhn-Tucker criteria and is completely polynomial in nature (being completely polynomial is not absolutely necessary to find solutions, but it is to find a global solution). One can then use the techniques of ...
What is algebraic geometry? - Mathematics Stack Exchange
2019年7月13日 · Algebraic geometry is about dealing with systems of polynomial equations. As you may imagine, this is much harder. In linear algebra, much of the theory is entirely independent of the field F, at least until you want to diagonalize operators; in algebraic geometry, non-algebraically-closed F are a massive headache, and there are phenomena in ...
algebraic geometry - How is Category Theory used to study …
2018年2月20日 · More generally, solution sets of polynomial equations (and more generally, algebraic varieties) are a central study object of algebraic geometry. As differential equations are central to all areas of physics, I assume that there have been made a lot of attempts to generalise these ideas to solution sets of these.
Famous papers in algebraic geometry - Mathematics Stack Exchange
2012年6月21日 · Serre's Faisceaux Algébriques Cohérents (=FAC) has the unique status of being: . a) Arguably the most important article in 20-th century algebraic geometry : it introduced sheaf-theoretic methods into algebraic geometry, including their cohomology, characterization of affine varieties by vanishing of said cohomology for coherent sheaves, twisting sheaves …