Affine Algebraic Geometry: Geometry of Polynomial Rings

Algebraic geometry is more advanced with the completeness condition for projective or complete varieties. Many geometric properties are well described by the finiteness or the vanishing of sheaf cohomologies on such varieties. For non-complete varieties like affine algebraic varieties, sheaf cohomology does not work well and research progress used to be slow, although affine spaces and polynomial rings are fundamental building blocks of algebraic geometry. Progress was rapid since the Abhyankar–Moh–Suzuki Theorem of embedded affine line was proved, and logarithmic geometry was introduced by Iitaka and Kawamata. Readers will find the book covers vast basic material on an extremely rigorous level: It begins with an introduction to algebraic geometry which comprises almost all results in commutative algebra and algebraic geometry. Arguments frequently used in affine algebraic geometry are elucidated by treating affine lines embedded in the affine plane and automorphism theorem of the affine plane. There is also a detailed explanation on affine algebraic surfaces which resemble the affine plane in the ring-theoretic nature and for actions of algebraic groups. The Jacobian conjecture for these surfaces is also considered by making use of the results and tools already presented in this book. The conjecture has been thought as one of the most unattackable problems even in dimension two. Advanced results are collected in appendices of chapters so that readers can understand the main streams of arguments. There are abundant problems in the first three chapters which come with hints and ideas for proof.