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Taking off from Euclid´s famous assertion to the contrary, this book leads the reader on a clear path - a royal road - through the elements of modern algebraic geometry from algebraic curves to the present shape of the field to Grothendieck´s theory of schemes.

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This book is about modern algebraic geometry. The title A Royal Road to Algebraic Geometry is inspired by the famous anecdote about the king asking Euclid if there really existed no simpler way for learning geometry, than to read all of his work Elements . Euclid is said to have answered: " There is no royal road to geometry !"

The book starts by explaining this enigmatic answer, the aim of the book being to argue that indeed, in some sense there is a royal road to algebraic geometry.

From a point of departure in algebraic curves, the exposition moves on to the present shape of the field, culminating with Alexander Grothendieck´s theory of schemes. Contemporary homological tools are explained.

The reader will follow a directed path leading up to the main elements of modern algebraic geometry. When the road is completed, the reader is empowered to start navigating in this immense field, and to open up the door to a wonderful field of research. The greatest scientific experience of a lifetime!

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Part I Curves: 1 Affine and Projective Space.- 2 Curves in A2 k and in P2.- 3 Higher Geometry in the Projective Plane.- 4 Plane Curves and Algebra.- 5 Projective Varieties in PNk.- Part II Introduction to Grothendieck´s Theory of Schemes: 6 Categories and Functors.- 7 Constructions and Representable Functors.- 8 Abelian Categories.- 9 The Concept of Spec(A).- 10 The Category of Schemes.- 11 Properties of Morphisms of Schemes.- 12 Modules, Algebras and Bundles on a Scheme.- 13 More Properties of Morphisms, Scheme Theoretic Image and the "Sorite".- 14 Projective Schemes and Bundles.- 15 Further Properties of Morphisms.- 16 Conormal Sheaf and Projective Bundles.- 17 Cohomology Theory on Schemes.- 18 Intersection Theory.- 19 Characteristic Classes in Algebraic Geometry.- 20 The Riemann-Roch Theorem.- 21 Some Basic constructions in the category of projective kvarieties.- 22 More on Duality.- References.- Index

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