Multiplier ideals in two-dimensional local rings with rational singularities. (English) Zbl 1357.14025
Consider an arbitrary ideal \(a\) on a regular complex surface germ \(X\), or, more generally, on the germ of a rational surface singularity \(X\). The main result in this paper is an efficient algorithm to compute sequentially the jumping numbers of \(a\), and also the whole chain of multiplier ideals of \(a\).
More precisely, the multiplier ideals \(O_X \supset J(a^\lambda), \lambda \in \mathbb Q_{>0},\) are well known (singularity) invariants of \(a\), satisfying \(J(a^{\lambda}) \supseteq J(a^{\lambda'})\) when \(\lambda < \lambda'\). The rational numbers \(0<\lambda_1 < \lambda_2 < \dots\) where these ideals change are the jumping numbers of \(a\). There is a known complete list of candidate jumping numbers in terms of the minimal log principalization \(\pi:Y\to X\) of \(a\). Write the (effective) divisor associated to the principal ideal \(a O_Y\) as \(\sum_i e_i E_i\), where \(E_i, i\in I,\) are the irreducible components of its support. Write also the relative canonical divisor \(K_\pi\) as \(\sum_i k_i E_i\). Then the jumping numbers of \(a\) are included in the list \(\frac {k_i +m}{e_i}\), where \(i\in I\) and \(m\in \mathbb Z_{>0}\).
K. Tucker [Trans. Am. Math. Soc. 362, No. 6, 3223–3241 (2010; Zbl 1194.14026)] developed an algorithm, deciding whether such a candidate is a jumping number or not, yielding then the list of actual jumping numbers. (This list is periodic, one only has to investigate the range \((0,2]\).) The authors proceed in a different and more efficient way; they construct an algorithm that computes any jumping number in terms of the previous one (thus avoiding \` candidates\'). Their main technical result to achieve this is as follows. For a given \(\lambda \in \mathbb Q_{>0}\), let \(D_\lambda= \sum_i e_i^\lambda E_i\) be the antinef closure of the round down of the (\(\mathbb Q\)-)divisor \(\lambda\text{div}(a O_Y) - K_\pi\). Then the jumping number consecutive to \(\lambda\) is the minimum of the numbers \(\frac{k_i+1+e_i^\lambda}{e_i}, i\in I\). In addition, any multiplier ideal is computed from the \` previous one\'. Important in the arguments is the new concept of minimal jumping divisor.
In the meantime, H. Baumers and F. Dachs-Cadefau extended some results of the present paper partly to higher dimensions [“Computing jumping numbers in higher dimensions”, arXiv:1603.00787].
More precisely, the multiplier ideals \(O_X \supset J(a^\lambda), \lambda \in \mathbb Q_{>0},\) are well known (singularity) invariants of \(a\), satisfying \(J(a^{\lambda}) \supseteq J(a^{\lambda'})\) when \(\lambda < \lambda'\). The rational numbers \(0<\lambda_1 < \lambda_2 < \dots\) where these ideals change are the jumping numbers of \(a\). There is a known complete list of candidate jumping numbers in terms of the minimal log principalization \(\pi:Y\to X\) of \(a\). Write the (effective) divisor associated to the principal ideal \(a O_Y\) as \(\sum_i e_i E_i\), where \(E_i, i\in I,\) are the irreducible components of its support. Write also the relative canonical divisor \(K_\pi\) as \(\sum_i k_i E_i\). Then the jumping numbers of \(a\) are included in the list \(\frac {k_i +m}{e_i}\), where \(i\in I\) and \(m\in \mathbb Z_{>0}\).
K. Tucker [Trans. Am. Math. Soc. 362, No. 6, 3223–3241 (2010; Zbl 1194.14026)] developed an algorithm, deciding whether such a candidate is a jumping number or not, yielding then the list of actual jumping numbers. (This list is periodic, one only has to investigate the range \((0,2]\).) The authors proceed in a different and more efficient way; they construct an algorithm that computes any jumping number in terms of the previous one (thus avoiding \` candidates\'). Their main technical result to achieve this is as follows. For a given \(\lambda \in \mathbb Q_{>0}\), let \(D_\lambda= \sum_i e_i^\lambda E_i\) be the antinef closure of the round down of the (\(\mathbb Q\)-)divisor \(\lambda\text{div}(a O_Y) - K_\pi\). Then the jumping number consecutive to \(\lambda\) is the minimum of the numbers \(\frac{k_i+1+e_i^\lambda}{e_i}, i\in I\). In addition, any multiplier ideal is computed from the \` previous one\'. Important in the arguments is the new concept of minimal jumping divisor.
In the meantime, H. Baumers and F. Dachs-Cadefau extended some results of the present paper partly to higher dimensions [“Computing jumping numbers in higher dimensions”, arXiv:1603.00787].
Reviewer: Wim Veys (Leuven)
Citations:
Zbl 1194.14026References:
| [1] | M. Alberich-Carramiñana, J. Àlvarez Montaner, and G. Blanco, Effective computation of base points of two-dimensional ideals , |
| [2] | M. Alberich-Carramiñana, J. Àlvarez Montaner, F. Dachs-Cadefau, and V. González-Alonso, Poincaré series of multiplier ideals in two-dimensional local rings with rational singularities , preprint, arXiv: |
| [3] | M. Artin, Some numerical criteria for contractability of curves on algebraic surfaces , Amer. J. Math. 84 (1962), 485-496. · Zbl 0105.14404 · doi:10.2307/2372985 |
| [4] | –, On isolated rational singularities of surfaces , Amer. J. Math. 88 (1966), 129-136. · Zbl 0142.18602 · doi:10.2307/2373050 |
| [5] | E. Casas-Alvero, Singularities of plane curves , London Math. Soc. Lecture Note Ser., 276, Cambridge University Press, Cambridge, 2000. · Zbl 0967.14018 |
| [6] | L. Ein, R. Lazarsfeld, K. Smith, and D. Varolin, Jumping coefficients of multiplier ideals , Duke Math. J. 123 (2004), 469-506. · Zbl 1061.14003 · doi:10.1215/S0012-7094-04-12333-4 |
| [7] | F. Enriques and O. Chisini, Lezioni sulla teoria geometrca delle equazioni e delle funzioni algebriche , N. Zanichelli, Bologna, 1915. · Zbl 0009.15904 |
| [8] | C. Favre and M. Jonsson, The valuative tree , Lecture Notes in Math., 1853, Springer-Verlag, Berlin, 2004. · Zbl 1064.14024 · doi:10.1007/b100262 |
| [9] | –, Valuations and multiplier ideals , J. Amer. Math. Soc. 18 (2005), 655-684. · Zbl 1075.14001 · doi:10.1090/S0894-0347-05-00481-9 |
| [10] | C. Galindo, F. Hernando, and F. Monserrat, The log-canonical threshold of a plane curve , Math. Proc. Cambridge Math. Soc. 160 (2016), 513-535. · Zbl 1371.14021 · doi:10.1017/S0305004116000037 |
| [11] | C. Galindo and F. Monserrat, The Poincaré series of multiplier ideals of a simple complete ideal in a local ring of a smooth surface , Adv. Math. 225 (2010), 1046-1068. · Zbl 1206.14011 · doi:10.1016/j.aim.2010.03.008 |
| [12] | D. Grayson and M. Stillman, Macaulay 2 , http://www.math.uiuc.edu/Macaulay2. URL: |
| [13] | G.-M. Greuel, G. Pfister, and H. Schönemann, Singular 2.0.5: A computer algebra system for polynomial computations , Centre for Computer Algebra, University of Kaiserslautern, 2003, http://www.singular.uni-kl.de. URL: |
| [14] | E. Hyry and T. Järviletho, Jumping numbers and ordered tree structures on the dual graph , Manuscripta Math. 136 (2011), 411-437. · Zbl 1235.13018 · doi:10.1007/s00229-011-0449-6 |
| [15] | T. Järviletho, Jumping numbers of a simple complete ideal in a two-dimensional regular local ring , Mem. Amer. Math. Soc. 214 (2011), no. 1009. |
| [16] | T. Kuwata, On log canonical thresholds of reducible plane curves , Amer. J. Math. 121 (1999), 701-721. · Zbl 0945.14002 · doi:10.1353/ajm.1999.0028 |
| [17] | H. Laufer, On rational singularities , Amer. J. Math. 94 (1972), 597-608. · Zbl 0251.32002 · doi:10.2307/2374639 |
| [18] | R. Lazarsfeld, Positivity in algebraic geometry II , 49, Springer-Verlag, Berlin, 2004. · Zbl 1093.14500 |
| [19] | J. Lipman, Rational singularities, with applications to algebraic surfaces and unique factorization , Inst. Hautes Études Sci. Publ. Math. 36 (1969), 195-279. · Zbl 0181.48903 · doi:10.1007/BF02684604 |
| [20] | J. Lipman and K. I. Watanabe, Integrally closed ideals in two-dimensional regular local rings are multiplier ideals , Math. Res. Lett. 10 (2003), 423-434. · Zbl 1049.13003 · doi:10.4310/MRL.2003.v10.n4.a2 |
| [21] | D. Naie, Jumping numbers of a unibranch curve on a smooth surface , Manuscripta Math. 128 (2009), 33-49. · Zbl 1165.14025 · doi:10.1007/s00229-008-0223-6 |
| [22] | A. J. Reguera, Curves and proximity on rational surface singularities , J. Pure Appl. Algebra 122 (1997), 107-126. · Zbl 0905.14018 · doi:10.1016/S0022-4049(96)00069-2 |
| [23] | K. E. Smith and H. Thompson, Irrelevant exceptional divisors for curves on a smooth surface , Algebra, geometry and their interactions, Contemp. Math., 448, pp. 245-254, Amer. Math. Soc., Providence, RI, 2007. · Zbl 1141.14004 · doi:10.1090/conm/448/08669 |
| [24] | K. Tucker, Integrally closed ideals on log terminal surfaces are multiplier ideals , Math. Res. Lett. 16 (2009), 903-908. · Zbl 1192.14016 · doi:10.4310/MRL.2009.v16.n5.a12 |
| [25] | –, Jumping numbers on algebraic surfaces with rational singularities , Trans. Amer. Math. Soc. 362 (2010), 3223-3241. · Zbl 1194.14026 · doi:10.1090/S0002-9947-09-04956-3 |
| [26] | W. Veys, Determination of the poles of the topological zeta function for curves , Manuscripta Math. 87 (1995), 435-448. · Zbl 0851.14012 · doi:10.1007/BF02570485 |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
