UPC: 9783658307325 | Springer Studium Mathematik – Master: Algebraic Geometry I: Schemes: With Examples and Exercises (Paperback)

UPC lookup results for: 9783658307325 | Springer Studium Mathematik – Master: Algebraic Geometry I: Schemes: With Examples and Exercises (Paperback)

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 .