Multidimensional continued fractions for cyclic quotient singularities and Dedekind sums

Hilbert desingularizations for three dimensional canonical cyclic quotient singularities

“…For the cyclic groups G in the cases (ii)-(β), (iii), and (iv) in the classification of three dimensional canonical cyclic quotient singularities in Section 1, the generator g of G is semi-unimodular if and only if Ashikaga's continued fraction consists of a round down polynomial and a remainder polynomial, and these polynomials are obtained via round down maps and remainder maps for a semi-unimodular proper fraction. For the details, see [1]. In this paper, the round down polynomials play important roles especially.…”

“…This expanding of Ashikaga's continued fraction indicates that the toric variety after the blow-up with the Oka center 1 11 (1, 2, 8) has two semi-isolated quotient singularities of 1 2 (1, 1, 0)type and 1 8 (1, 2, 5)-type. For these quotient singularities, the corresponding cones which appear in σ after the subdivision by 1 11 (1, 2, 8) ∈ N are σ 2 = R ≥0 e 1 + R ≥0 c + R ≥0 e 3 and σ 3 = R ≥0 e 1 + R ≥0 e 2 + R ≥0 c respectively. 1 2 (1, 1, 0) and 1 8 (1, 2, 5) are the Oka center of semiunimodular cones σ 2 , σ 3 over e 1 respectively.…”

“…For these quotient singularities, the corresponding cones which appear in σ after the subdivision by 1 11 (1, 2, 8) ∈ N are σ 2 = R ≥0 e 1 + R ≥0 c + R ≥0 e 3 and σ 3 = R ≥0 e 1 + R ≥0 e 2 + R ≥0 c respectively. 1 2 (1, 1, 0) and 1 8 (1, 2, 5) are the Oka center of semiunimodular cones σ 2 , σ 3 over e 1 respectively. Therefore, we can take blow-ups with the Oka centers again.…”

“…Therefore, we can take blow-ups with the Oka centers again. The blow-up with Oka centers of X(N , σ 3 ) consists a smooth toric variety and quotient singularities of 1 2 (1, 0, 1)-type and 1 5 (1, 2, 2)-type respectively. By repeating blow-ups with Oka centers, we have the smooth toric variety (see Fig.…”

“…r (a 1 , a 2 , a 3 ) with gcd(a i , a j ) = 1 and a i + a j = r for some 1 ≤ i < j ≤ 3, (iii) G = 1 4k (1, 2k + 1, 4k − 2) with k ≥ 2, (iv) G = 1 9 (1, 4, 7) or 1 14 (1,9,11). The class (ii) has the two cases The form (ii)-(α) is a symbolic form as a three dimensional toric terminal quotient singularity.…”