Some remarks on homogeneous minimal reductions

Let I be a homogeneous ideal of a graded affine k–algebra R such that there exists some homogeneous minimal reduction. We prove that the degrees (of a basis) of every homogeneous minimal reduction J of I are uniquely determined by I; moreover if the fiber cone F(I) is reduced, then the last degree of J is equal to the last degree of I. Moreover, if R is Cohen–Macaulay and I is of analytic deviation one, with 0 < ht(I) := g, it is shown that the first g degrees of J are equals to the first g degrees of I. These results are applied to the ideals I of k[x_0, . . . , x_{d−1}], whichhave scheme–theoretic generations of length ht(I) + 2. Some examples are given.