Overview
- Authors:
-
-
Ulrich Görtz
-
Institute of Experimental Mathematics, University Duisburg-Essen, Essen, Germany
-
Torsten Wedhorn
-
University of Paderborn, Department of Mathematics, Paderborn, Germany
- Der Wegbegleiter in das Feld der modernen
- algebraischen Geometrie im Bachelor/Master Studium
Access this book
Other ways to access
Table of contents (17 chapters)
-
-
- Ulrich Görtz, Torsten Wedhorn
Pages 1-6
-
- Ulrich Görtz, Torsten Wedhorn
Pages 7-39
-
- Ulrich Görtz, Torsten Wedhorn
Pages 40-65
-
- Ulrich Görtz, Torsten Wedhorn
Pages 66-92
-
- Ulrich Görtz, Torsten Wedhorn
Pages 93-117
-
- Ulrich Görtz, Torsten Wedhorn
Pages 118-144
-
- Ulrich Görtz, Torsten Wedhorn
Pages 145-168
-
- Ulrich Görtz, Torsten Wedhorn
Pages 169-204
-
- Ulrich Görtz, Torsten Wedhorn
Pages 205-225
-
- Ulrich Görtz, Torsten Wedhorn
Pages 226-240
-
- Ulrich Görtz, Torsten Wedhorn
Pages 241-285
-
- Ulrich Görtz, Torsten Wedhorn
Pages 286-319
-
- Ulrich Görtz, Torsten Wedhorn
Pages 320-365
-
- Ulrich Görtz, Torsten Wedhorn
Pages 366-422
-
- Ulrich Görtz, Torsten Wedhorn
Pages 423-484
-
- Ulrich Görtz, Torsten Wedhorn
Pages 485-502
-
- Ulrich Görtz, Torsten Wedhorn
Pages 503-540
-
Back Matter
Pages 541-615
About this book
Algebraic geometry has its origin in the study of systems of polynomial equations f (x ,. . . ,x )=0, 1 1 n . . . f (x ,. . . ,x )=0. r 1 n Here the f ? k[X ,. . . ,X ] are polynomials in n variables with coe?cients in a ?eld k. i 1 n n ThesetofsolutionsisasubsetV(f ,. . . ,f)ofk . Polynomialequationsareomnipresent 1 r inandoutsidemathematics,andhavebeenstudiedsinceantiquity. Thefocusofalgebraic geometry is studying the geometric structure of their solution sets. n If the polynomials f are linear, then V(f ,. . . ,f ) is a subvector space of k. Its i 1 r “size” is measured by its dimension and it can be described as the kernel of the linear n r map k ? k , x=(x ,. . . ,x ) ? (f (x),. . . ,f (x)). 1 n 1 r For arbitrary polynomials, V(f ,. . . ,f ) is in general not a subvector space. To study 1 r it, one uses the close connection of geometry and algebra which is a key property of algebraic geometry, and whose ?rst manifestation is the following: If g = g f +. . . g f 1 1 r r is a linear combination of the f (with coe?cients g ? k[T ,. . . ,T ]), then we have i i 1 n V(f ,. . . ,f)= V(g,f ,. . . ,f ). Thus the set of solutions depends only on the ideal 1 r 1 r a? k[T ,. . . ,T ] generated by the f .
Authors and Affiliations
-
Institute of Experimental Mathematics, University Duisburg-Essen, Essen, Germany
Ulrich Görtz
-
University of Paderborn, Department of Mathematics, Paderborn, Germany
Torsten Wedhorn
About the authors
Prof. Dr. Ulrich Görtz, Institut für Experimentelle Mathematik, Universität Duisburg-Essen.Essen.
Prof. Dr. Torsten Wedhorn, Institut für Mathematik, Universität Paderborn.
Prof. Dr. Ulrich Görtz, Institute of Experimental Mathematics, University Duisburg-Essen.
Prof. Dr. Torsten Wedhorn, Department of Mathematics, University of Paderborn.