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The primary audience for this book is students and the young
researchers interested in the core of the discipline. Commutative
algebra is by and large a self-contained discipline, which makes it
quite dry for the beginner with a basic training in elementary
algebra and calculus. A stable mathematical discipline such as this
enshrines a vital number of topics to be learned at an early stage,
more or less universally accepted and practiced. Naturally, authors
tend to turn these topics into an increasingly short and elegant
list of basic facts of the theory. So, the shorter the better.
However, there is a subtle watershed between elegance and
usefulness, especially if the target is the beginner. From my
experience throughout years of teaching, elegance and terseness do
not do it, except much later in the carrier. To become useful, the
material ought to carry quite a bit of motivation through
justification and usefulness pointers. On the other hand, it is
difficult to contemplate these teaching devices in the writing of a
short book. I have divided the material in three parts. starting
with more elementary sections, then carrying an intermezzo on more
difficult themes to make up for a smooth crescendo with additional
tools and, finally, the more advanced part, versing on a reasonable
chunk of present-day steering of commutative algebra. Historic
notes at the end of each chapter provide insight into the original
sources and background information on a particular subject or
theorem. Exercises are provided and propose problems that apply the
theory to solve concrete questions (yes, with concrete polynomials,
and so forth).
The objective of this book is to look at certain commutative graded
algebras that appear frequently in algebraic geometry. By studying
classical constructions from geometry from the point of view of
modern commutative algebra, this carefully-written book is a
valuable source of information, offering a careful algebraic
systematization and treatment of the problems at hand, and
contributing to the study of the original geometric questions. In
greater detail, the material covers aspects of rational maps
(graph, degree, birationality, specialization, combinatorics),
Cremona transformations, polar maps, Gauss maps, the geometry of
Fitting ideals, tangent varieties, joins and secants, Aluffi
algebras. The book includes sections of exercises to help put in
practice the theoretic material instead of the mere complementary
additions to the theory.
The central theme of this volume is commutative algebra, with
emphasis on special graded algebras, which are increasingly of
interest in problems of algebraic geometry, combinatorics and
computer algebra. Most of the papers have partly survey character,
but are research-oriented, aiming at classification and structural
results.
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