This book is an introduction to the geometry of complex algebraic
varieties. It is intended for students who have learned algebra,
analysis, and topology, as taught in standard undergraduate
courses. So it is a suitable text for a beginning graduate course
or an advanced undergraduate course. The book begins with a study
of plane algebraic curves, then introduces affine and projective
varieties, going on to dimension and construcibility.
$\mathcal{O}$-modules (quasicoherent sheaves) are defined without
reference to sheaf theory, and their cohomology is defined
axiomatically. The Riemann-Roch Theorem for curves is proved using
projection to the projective line. Some of the points that aren't
always treated in beginning courses are Hensel's Lemma, Chevalley's
Finiteness Theorem, and the Birkhoff-Grothendieck Theorem. The book
contains extensive discussions of finite group actions, lines in
$\mathbb{P}^3$, and double planes, and it ends with applications of
the Riemann-Roch Theorem.
General
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