University of Paderborn Department of Mathematics Diploma Thesis ...

Glossary of NotationsK a field of characteristic zero, 7Rthe polynomial ring K[x 0 , . . . , x n ] in n + 1 variables over the fieldK, 7deg f the degree of the polynomial f, 7R(−k) shift of the grading by −k, 7M the set of monomials in the polynomial ring R := K[x 0 , . . . , x n ], 7[R] d vector space of homogeneous polynomials f of degree d, 8dim K [R] d K-vector space dimension, 8µ(I) number of minimal generators of the ideal I, 8Rad I the radical of the ideal I, 9gcd(f, g) greatest common divisor of the polynomials f and g, 9[I] d vector space of homogeneous polynomials f of degree d in I, 9m the unique homogeneous maximal ideal (x 0 , . . . , x n ), 10sat xn (I) saturation of the ideal I in K[x 0 , . . . , x n ], 10h I Hilbert function of the ideal I, 10h R/I Hilbert function of R/I, 11p I Hilbert polynomial of the ideal I, 11p R/I Hilbert polynomial of R/I, 11H I Hilbert series of the ideal I, 11H R/I Hilbert series of R/I, 11a 〈d〉 16GBGL(n + 1, K), i.e. the multiplicative group of all invertible(n + 1) × (n + 1)-matrices over K, 19the Borel-subgroup of G, i.e. the invertible upper triangular(n + 1) × (n + 1)-matrices over K, 20127