Abstract
This chapter discusses several basic notions of algebraic geometry and their connection with commutative algebra, especially the connection with polynomial rings. The chapter reviews results that are related to Noether's normalization theorem. Although it is important to study polynomial rings without any restriction on the coefficient ring k, it is also true that the most basic case occurs when k is a field. There are difficult problems that are caused by the simplicity of the structure of polynomial rings. As an example of the complexity of phenomena related to polynomial rings, or equivalently to rational varieties, the chapter discusses the birational correspondences of ruled surfaces. The discussion is related to the automorphism groups of polynomial rings and Cremona groups.
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