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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.
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.)
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.
Foundations of Algebraic Geometry math216.wordpress.com November 18, 2017 draft ⃝c 2010–2017 by Ravi Vakil. Note to reader: the index and formatting have yet to be properly dealt with. There remain many issues still to be dealt with in the main part of the notes (including many of your corrections and suggestions).
We will get an understanding of the geometry of a plane curve as we go along, and we mention just one point here. A plane curve is called a curve because it is defined by one equation in two variables. Its algebraic dimension is one. But because our scalars are complex numbers, it will be a surface, geometrically. This is
The basic objects of investigation in algebraic geometry are so-called algebraic varieties, 1 which are (roughly speaking) the geometric point-sets that can be cut out by systems of algebraic equations.
To Hema, Ashok, and Maya. Preface. This book is an introductory course in algebraic geometry, proving most of the fundamental classical results of algebraic geometry. Algebraic geometry combines the intuition of geometry with the preci-sion of algebra.
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.
It should be clear, therefore, that any brief introduction to algebraic ge-ometry has to be selective and can at best hope to provide some glimpses of the subject. This is what we have set out to do. In fact, we will fo-cus mainly on two basic results in algebraic geometry, known as Bezout’s
