An ℓ-th root of a test configuration of exponent ℓ

Let (X, L) be a polarized algebraic manifold. Then for every test configuration μ = (X, L,Ψ) for (X, L) of exponent ℓ, we obtain an ℓ-th root (κ, D) of μ and Gm-equivariant desingularizations ι : ^X → X and η : ^X → Y, both isomorphic on^X \^X 0, such thatwhereκ= (Y, Q, η) is a test configuration for (X, L) of exponent 1, and D is an effective Q-divisor on^X such that ℓD is an integral divisor with support in the fiber X0. Then (κ, D) can be chosen in such a way thatwhere C1 and C2 are positive real constants independent of the choice of μ and ℓ. This plays an important role in our forthcoming papers on the existence of constant scalar curvature Kähler metrics (cf. [6]) and also on the compactified moduli space of test configurations (cf. [5],[7]).

High Common Factor

As the colleges now are requiring highest common factor by factoring methods only, any plan whereby the number of factors to be tried can be lessened is certainly worth while. For in general we must regard as possible any binomial factor whose first-degree term is an integral divisor of the highest term of the given expression and whose independent term is an integral divisor of the independent term of the expression. Thus in such a problem as, “Find the H.C.F. of 5x3 − 21x2 + 5x − 4 and 5x3 − 19x2 + 5x + 4” (McCurdy’s Exercise Book, page 40, example 4) the possible factors that a student might try and must try, unless he were very lucky in those he chose to try first, are x ± 1, x ± 2, x ± 4, 5x ± 1, 5x ± 2, 5x ± 4. And further if the given expressions were, say, cubics having no common binomial factor at all but with a quadratic one instead (e. g., 2x3 + 5x2 + x − 3 and 2x3 − x2 − 5x + 3) I doubt if the ordinary first-year student would get any result unless it were unity.