Browse > Article
http://dx.doi.org/10.4134/JKMS.j190739

A GRADED MINIMAL FREE RESOLUTION OF THE m-TH ORDER SYMBOLIC POWER OF A STAR CONFIGURATION IN ℙn  

Park, Jung Pil (Faculty of Liberal Education Seoul National University)
Shin, Yong-Su (Department of Mathematics Sungshin Women's University)
Publication Information
Journal of the Korean Mathematical Society / v.58, no.2, 2021 , pp. 283-308 More about this Journal
Abstract
In [30] the author finds a graded minimal free resolution of the 2-nd order symbolic power of a star configuration in ℙn of any codimension r. In this paper, we find that of any m-th order symbolic power of a star configuration in ℙn of codimension 2, which generalizes the result of Galetto, Geramita, Shin, and Van Tuyl in [15, Theorem 5.3]. Furthermore, we extend it to the m-th order symbolic power of a star configuration in ℙn of any codimension r for m = 3, 4, which also generalizes the result of Biermann et al. in [1, Corollaries 4.6 and 5.7]. We also suggest how to find a graded minimal free resolution of the m-th order symbolic power of a star configuration in ℙn of any codimension r for m ≥ 5.
Keywords
Symbolic powers; regular powers; star configurations; a graded minimal free resolution;
Citations & Related Records
연도 인용수 순위
  • Reference
1 M. Dumnicki, B. Harbourne, T. Szemberg, and H. Tutaj-Gasinska, Linear subspaces, symbolic powers and Nagata type conjectures, Adv. Math. 252 (2014), 471-491. https://doi.org/10.1016/j.aim.2013.10.029   DOI
2 L. Ein, R. Lazarsfeld, and K. E. Smith, Uniform bounds and symbolic powers on smooth varieties, Invent. Math. 144 (2001), no. 2, 241-252. https://doi.org/10.1007/s002220100121   DOI
3 S. Franceschini and A. Lorenzini, Fat points of ℙn whose support is contained in a linear proper subspace, J. Pure Appl. Algebra 160 (2001), no. 2-3, 169-182. https://doi.org/10.1016/S0022-4049(00)00084-0   DOI
4 F. Galetto, A. V. Geramita, Y. S. Shin, and A. Van Tuyl, The symbolic defect of an ideal, J. Pure Appl. Algebra 223 (2019), no. 6, 2709-2731. https://doi.org/10.1016/j.jpaa.2018.11.019   DOI
5 J. Biermann, H. de Alba, F. Galetto, S. Murai, U. Nagel, A. O'Keefe, T. Romer, and A. Seceleanu, Betti numbers of symmetric shifted ideals, J. Algebra 560 (2020), 312-342. https://doi.org/10.1016/j.jalgebra.2020.04.037   DOI
6 M. Hochster and C. Huneke, Comparison of symbolic and ordinary powers of ideals, Invent. Math. 147 (2002), no. 2, 349-369. https://doi.org/10.1007/s002220100176   DOI
7 M. Lampa-Baczynska and G. Malara, On the containment hierarchy for simplicial ideals, J. Pure Appl. Algebra 219 (2015), no. 12, 5402-5412. https://doi.org/10.1016/j.jpaa.2015.05.022   DOI
8 M. Mosakhani and H. Haghighi, On the configurations of points in ℙ2 with the Waldschmidt constant equal to two, J. Pure Appl. Algebra 220 (2016), no. 12, 3821-3825. https://doi.org/10.1016/j.jpaa.2016.05.014   DOI
9 Y. S. Shin, Secants to the variety of completely reducible forms and the Hilbert function of the union of star-configurations, J. Algebra Appl. 11 (2012), no. 6, 1250109, 27 pp. https://doi.org/10.1142/S0219498812501095   DOI
10 J. P. Park and Y. S. Shin, The minimal free graded resolution of a star-configuration in ℙn, J. Pure Appl. Algebra 219 (2015), no. 6, 2124-2133. https://doi.org/10.1016/j.jpaa.2014.07.026   DOI
11 Y. S. Shin, Star-configurations in ℙ2 having generic Hilbert function and the weak Lefschetz property, Comm. Algebra 40 (2012), no. 6, 2226-2242. https://doi.org/10.1080/00927872.2012.656783   DOI
12 C. Bocci and B. Harbourne, The resurgence of ideals of points and the containment problem, Proc. Amer. Math. Soc. 138 (2010), no. 4, 1175-1190. https://doi.org/10.1090/S0002-9939-09-10108-9   DOI
13 Y. S. Shin, A graded minimal free resolution of the 2nd order symbolic power of the ideal of a star configuration in ℙn, J. Korean Math. Soc. 56 (2019), no. 1, 169-181. https://doi.org/10.4134/JKMS.j180119   DOI
14 C. Bocci, S. Cooper, E. Guardo, B. Harbourne, M. Janssen, U. Nagel, A. Seceleanu, A. Van Tuyl, and T. Vu, The Waldschmidt constant for squarefree monomial ideals, J. Algebraic Combin. 44 (2016), no. 4, 875-904. https://doi.org/10.1007/s10801-016-0693-7   DOI
15 C. Bocci and B. Harbourne, Comparing powers and symbolic powers of ideals, J. Algebraic Geom. 19 (2010), no. 3, 399-417. https://doi.org/10.1090/S1056-3911-09-00530-X   DOI
16 F. Galetto, Y. S. Shin, and A. Van Tuyl, Distinguishing k-configurations, Illinois J. Math. 61 (2017), no. 3-4, 415-441. https://projecteuclid.org/euclid.ijm/1534924834   DOI
17 E. Carlini, E. Guardo, and A. Van Tuyl, Star configurations on generic hypersurfaces, J. Algebra 407 (2014), 1-20. https://doi.org/10.1016/j.jalgebra.2014.02.013   DOI
18 E. Carlini and A. Van Tuyl, Star configuration points and generic plane curves, Proc. Amer. Math. Soc. 139 (2011), no. 12, 4181-4192. https://doi.org/10.1090/S0002-9939-2011-11204-8   DOI
19 O. Zariski and P. Samuel, Commutative Algebra. Vol. II, The University Series in Higher Mathematics, D. Van Nostrand Co., Inc., Princeton, NJ, 1960.
20 A. V. Geramita, B. Harbourne, and J. Migliore, Star configurations in ℙn, J. Algebra 376 (2013), 279-299. https://doi.org/10.1016/j.jalgebra.2012.11.034   DOI
21 B. Harbourne, The ideal generation problem for fat points, J. Pure Appl. Algebra 145 (2000), no. 2, 165-182. https://doi.org/10.1016/S0022-4049(98)00077-2   DOI
22 A. V. Geramita, B. Harbourne, J. Migliore, and U. Nagel, Matroid configurations and symbolic powers of their ideals, Trans. Amer. Math. Soc. 369 (2017), no. 10, 7049-7066. https://doi.org/10.1090/tran/6874   DOI
23 A. V. Geramita, J. Migliore, and L. Sabourin, On the first infinitesimal neighborhood of a linear configuration of points in ℙ2, J. Algebra 298 (2006), no. 2, 563-611. https://doi.org/10.1016/j.jalgebra.2006.01.035   DOI
24 E. Guardo and A. Van Tuyl, Powers of complete intersections: graded Betti numbers and applications, Illinois J. Math. 49 (2005), no. 1, 265-279. http://projecteuclid.org/euclid.ijm/1258138318   DOI
25 S. Cooper, B. Harbourne, and Z. Teitler, Combinatorial bounds on Hilbert functions of fat points in projective space, J. Pure Appl. Algebra 215 (2011), no. 9, 2165-2179. https://doi.org/10.1016/j.jpaa.2010.12.006   DOI
26 B. Harbourne and C. Huneke, Are symbolic powers highly evolved?, J. Ramanujan Math. Soc. 28A (2013), 247-266.
27 J. Herzog and Y. Takayama, Resolutions by mapping cones, Homology Homotopy Appl. 4 (2002), no. 2, part 2, 277-294. https://doi.org/10.4310/hha.2002.v4.n2.a13   DOI
28 M. V. Catalisano, A. V. Geramita, A. Gimigliano, B. Harbourne, J. Migliore, U. Nagel, and Y. S. Shin, Secant varieties of the varieties of reducible hypersurfaces in ℙn, J. Algebra 528 (2019), 381-438. https://doi.org/10.1016/j.jalgebra.2019.03.014   DOI
29 M. V. Catalisano, A. V. Geramita, A. Gimigliano, and Y. S. Shin, The secant line variety to the varieties of reducible plane curves, Ann. Mat. Pura Appl. (4) 195 (2016), no. 2, 423-443. https://doi.org/10.1007/s10231-014-0470-y   DOI
30 M. V. Catalisano, E. Guardo, and Y. S. Shin, The Waldschmidt constant of special k-configurations in ℙn, J. Pure Appl. Algebra 224 (2020), no. 10, 106341, 28 pp. https://doi.org/10.1016/j.jpaa.2020.106341   DOI
31 M. DiPasquale, C. A. Francisco, J. Mermin, and J. Schweig, Asymptotic resurgence via integral closures, Trans. Amer. Math. Soc. 372 (2019), no. 9, 6655-6676. https://doi.org/10.1090/tran/7835   DOI