Plane curves were the first algebraic varieties to be studied, so we begin with them. They provide helpful examples, and we will see in Chapter 5 how they control varieties of arbitrary dimension.
The main objects of study in algebraic geometry are systems of algebraic equa-tions and their sets of solutions. Let kbe a eld and k[T 1;:::;T n] = k[T] be the algebra of polynomials in nvariables over k. A system of algebraic equations over kis an expression fF= 0g F2S; where Sis a subset of k[T]. We shall often identify it with the subset S.
a -algebra and -module ,weoftenwrite for ⊗ .ThedualHom -linear( , ) ofa -vectorspace isdenotedby ∨ .Allringsarecommutativewith1,andhomomor- phismsarerequiredtomap1to1.FollowingBourbaki,werequirecompacttopological
The strict definition of the algebraic geometry is the study of solutions of polynomial equations. But very rarely equations are explicitly written in a problem one may solve.
Algebraic geometry is a beautiful subject, and it’s usually taught as a mid-level graduate course, so we’ll need to discuss things in this class without a lot of background. In particular, we won’t assume commutative algebra (18.705), though that might be useful. (18.702 is essential, though.)
0 Algebraic geometry Algebraic geometry is the study of algebraic varieties: objects which are the zero locus of a polynomial or several polynomials. One might argue that the discipline goes back to Descartes. Many mathematicians—such as Abel, Riemann, Poincar´e, M. Noether, Severi’s Italian school, and more recently
The prerequisites are standard undergraduate courses in algebra, analysis, and topology, and the definitions of category and functor. I also suppose a familiarity with the implicit function theorem for complex variables.
Early (out-of-date) version of The Rising Sea: Foundations of Algebraic Geometry (c) 2024 Ravi Vakil. Published by Princeton University Press.
Early (out-of-date) version of The Rising Sea: Foundations of Algebraic Geometry (c) 2024 Ravi Vakil. Published by Princeton University Press.
Lecture Notes | Algebraic Geometry - MIT OpenCourseWare
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