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This book introduces the reader to modern algebraic geometry. It
presents Grothendieck's technically demanding language of schemes
that is the basis of the most important developments in the last
fifty years within this area. A systematic treatment and motivation
of the theory is emphasized, using concrete examples to illustrate
its usefulness. Several examples from the realm of Hilbert modular
surfaces and of determinantal varieties are used methodically to
discuss the covered techniques. Thus the reader experiences that
the further development of the theory yields an ever better
understanding of these fascinating objects. The text is
complemented by many exercises that serve to check the
comprehension of the text, treat further examples, or give an
outlook on further results. The volume at hand is an introduction
to schemes. To get startet, it requires only basic knowledge in
abstract algebra and topology. Essential facts from commutative
algebra are assembled in an appendix. It will be complemented by a
second volume on the cohomology of schemes.
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