Bounds of the multiplicity of abelian quotient complete intersection singularities
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© Springer-Verlag GmbH Germany, part of Springer Nature 2020
Kohsuke Shibata
Bounds of the multiplicity of abelian quotient complete intersection singularities Received: 9 August 2019 / Accepted: 30 October 2020 Abstract. In this paper we investigate the multiplicity and the log canonical threshold of abelian quotient complete intersection singularities in terms of the notion of special datum. Moreover we give bounds of the multiplicity of abelian quotient complete intersection singularities.
1. Introduction In [10], Watanabe classified all abelian quotient complete intersection singularities. Watanabe introduced the notion of special datum (see Sect. 2 for detailed definitions) in order to classify abelian quotient complete intersection singularities. Using a special datum, he gave an upper bound of the multiplicity of abelian quotient complete intersection singularities. Theorem 1.1. (Proposition 3.1 in [10]) Let G be a finite abelian subgroup of SL(n, C). If R = C[x1 , . . . , xn ]G is a complete intersection, then e(RmG ) ≤ 2n−1 , where mG = (x1 , . . . , xn ) ∩ R and e(RmG ) is the Hilbert–Samuel multiplicity of the local ring RmG . The following conjecture was posed by Watanabe as a generalization of Theorem 1.1. Conjecture 1.2. Let X be an n-dimensional variety which is locally a complete intersection with canonical singularities. Then e(O X,x ) ≤ 2n−1 for all closed points x of X . In [9], this conjecture was refined as follows: K. Shibata (B): Department of Mathematics, College of Humanities and Sciences, Nihon University, 3-25-40 Sakurajosui, Setagaya-Ku, Tokyo 156-8550, Japan. e-mail: [email protected] Mathematics Subject Classification: Primary 13H15; Secondary 14B05
https://doi.org/10.1007/s00229-020-01261-8
K. Shibata
Conjecture 1.3. Let X be an n-dimensional variety which is locally a complete intersection with log canonical singularities. Then e(O X,x ) ≤ 2n−lct(mx ) for all closed points x of X and equality holds if and only if emb(X, x) = 2n − lct(mx ), where emb(X, x) is the embedding dimension of X at x and lct(mx ) is the log canonical threshold of mx . In [9], the author gave upper bounds of the multiplicity by functions of the log canonical threshold for locally a complete intersection singularity. As an application, we obtained the affirmative answer to the conjecture if the dimension of the variety is less than or equal to 32 and the variety has canonical singularities. Theorem 1.4. (Theorem 5.6 in [9]) Let X be an n-dimensional variety of locally a complete intersection with canonical singularities. If n ≤ 32, then Conjecture 1.3 holds. In particular, if n ≤ 32, then Conjecture 1.2 holds. In this paper, we study the multiplicities and the log canonical thresholds of abelian quotient complete intersection singularities in terms of the special datum. We give an affirmative answer to Conjecture 1.3 for abelian quotient complete intersection singularities using Watanabe’s classification. Theorem 1.5. Let G be a finite abelian subgroup of SL(n, C). If R = C[x1 , . . . , xn ]G
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