Algebraic Geometry I: Schemes: With Examples and Exercises Grtz Ulrich
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…
Specifikacia Algebraic Geometry I: Schemes: With Examples and Exercises Grtz Ulrich
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