Algebraic Geometry

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 .