Interpolation and the weak Lefschetz property

Nagel, Uwe; Trok, Bill

doi:10.1090/tran/7889

Interpolation and the weak Lefschetz property
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by Uwe Nagel and Bill Trok PDF

Trans. Amer. Math. Soc. 372 (2019), 8849-8870 Request permission

Abstract:

Our starting point is a basic problem in Hermite interpolation theory—namely, determining the least degree of a homogeneous polynomial that vanishes to some specified order at every point of a given finite set. We solve this problem in many cases if the number of points is small compared to the dimension of their linear span. This also allows us to establish results on the Hilbert function of ideals generated by powers of linear forms. The Verlinde formula determines such a Hilbert function in a specific instance. We complement this result and also determine the Castelnuovo–Mumford regularity of the corresponding ideals. As applications, we establish new instances of conjectures by Chudnovsky and by Demailly on the Waldschmidt constant. Moreover, we show that conjectures on the failure of the weak Lefschetz property by Harbourne, Schenck, and Seceleanu as well as by Migliore, Miró-Roig, and the first author are true asymptotically. The latter also relies on a new result for Eulerian numbers.

References

Federico Ardila and Alexander Postnikov, Combinatorics and geometry of power ideals, Trans. Amer. Math. Soc. 362 (2010), no. 8, 4357–4384. MR 2608410, DOI 10.1090/S0002-9947-10-05018-X
Cristiano Bocci and Brian Harbourne, Comparing powers and symbolic powers of ideals, J. Algebraic Geom. 19 (2010), no. 3, 399–417. MR 2629595, DOI 10.1090/S1056-3911-09-00530-X
Cristiano Bocci, Susan Cooper, Elena Guardo, Brian Harbourne, Mike Janssen, Uwe Nagel, Alexandra Seceleanu, Adam Van Tuyl, and Thanh Vu, The Waldschmidt constant for squarefree monomial ideals, J. Algebraic Combin. 44 (2016), no. 4, 875–904. MR 3566223, DOI 10.1007/s10801-016-0693-7
G. V. Chudnovsky, Singular points on complex hypersurfaces and multidimensional Schwarz lemma, Seminar on Number Theory, Paris 1979–80, Progr. Math., vol. 12, Birkhäuser, Boston, Mass., 1981, pp. 29–69. MR 633888
David Cook II, The Lefschetz properties of monomial complete intersections in positive characteristic, J. Algebra 369 (2012), 42–58. MR 2959785, DOI 10.1016/j.jalgebra.2012.07.015
J.-P. Demailly, Formules de Jensen en plusieurs variables et applications arithmétiques, Bull. Soc. Math. France 110 (1982), no. 1, 75–102 (French, with English summary). MR 662130, DOI 10.24033/bsmf.1954
Roberta Di Gennaro, Giovanna Ilardi, and Jean Vallès, Singular hypersurfaces characterizing the Lefschetz properties, J. Lond. Math. Soc. (2) 89 (2014), no. 1, 194–212. MR 3174740, DOI 10.1112/jlms/jdt053
Marcin Dumnicki, An algorithm to bound the regularity and nonemptiness of linear systems in $\Bbb P^n$, J. Symbolic Comput. 44 (2009), no. 10, 1448–1462. MR 2543429, DOI 10.1016/j.jsc.2009.04.005
Marcin Dumnicki, Brian Harbourne, Tomasz Szemberg, and Halszka Tutaj-Gasińska, Linear subspaces, symbolic powers and Nagata type conjectures, Adv. Math. 252 (2014), 471–491. MR 3144238, DOI 10.1016/j.aim.2013.10.029
Marcin Dumnicki and Halszka Tutaj-Gasińska, A containment result in $P^n$ and the Chudnovsky conjecture, Proc. Amer. Math. Soc. 145 (2017), no. 9, 3689–3694. MR 3665024, DOI 10.1090/proc/13582
Lawrence Ein, Robert Lazarsfeld, and Karen E. Smith, Uniform bounds and symbolic powers on smooth varieties, Invent. Math. 144 (2001), no. 2, 241–252. MR 1826369, DOI 10.1007/s002220100121
David Eisenbud, The geometry of syzygies, Graduate Texts in Mathematics, vol. 229, Springer-Verlag, New York, 2005. A second course in commutative algebra and algebraic geometry. MR 2103875
J. Emsalem and A. Iarrobino, Inverse system of a symbolic power. I, J. Algebra 174 (1995), no. 3, 1080–1090. MR 1337186, DOI 10.1006/jabr.1995.1168
Hélène Esnault and Eckart Viehweg, Sur une minoration du degré d’hypersurfaces s’annulant en certains points, Math. Ann. 263 (1983), no. 1, 75–86 (French). MR 697332, DOI 10.1007/BF01457085
Louiza Fouli, Paolo Mantero, and Yu Xie, Chudnovsky’s conjecture for very general points in $\Bbb P^N_k$, J. Algebra 498 (2018), 211–227. MR 3754412, DOI 10.1016/j.jalgebra.2017.11.002
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. MR 3683102, DOI 10.1090/tran/6874
Elena Guardo, Brian Harbourne, and Adam Van Tuyl, Asymptotic resurgences for ideals of positive dimensional subschemes of projective space, Adv. Math. 246 (2013), 114–127. MR 3091802, DOI 10.1016/j.aim.2013.05.027
Brian Harbourne and Craig Huneke, Are symbolic powers highly evolved?, J. Ramanujan Math. Soc. 28A (2013), 247–266. MR 3115195
Brian Harbourne, Hal Schenck, and Alexandra Seceleanu, Inverse systems, Gelfand-Tsetlin patterns and the weak Lefschetz property, J. Lond. Math. Soc. (2) 84 (2011), no. 3, 712–730. MR 2855798, DOI 10.1112/jlms/jdr033
Tadahito Harima, Juan C. Migliore, Uwe Nagel, and Junzo Watanabe, The weak and strong Lefschetz properties for Artinian $K$-algebras, J. Algebra 262 (2003), no. 1, 99–126. MR 1970804, DOI 10.1016/S0021-8693(03)00038-3
Melvin Hochster and Craig Huneke, Comparison of symbolic and ordinary powers of ideals, Invent. Math. 147 (2002), no. 2, 349–369. MR 1881923, DOI 10.1007/s002220100176
Tian-Xiao He, Eulerian polynomials and B-splines, J. Comput. Appl. Math. 236 (2012), no. 15, 3763–3773. MR 2923509, DOI 10.1016/j.cam.2011.10.013
Le Tuan Hoa, Jürgen Stückrad, and Wolfgang Vogel, Towards a structure theory for projective varieties of $\textrm {degree}=\textrm {codimension}+2$, J. Pure Appl. Algebra 71 (1991), no. 2-3, 203–231. MR 1117635, DOI 10.1016/0022-4049(91)90148-U
A. Iarrobino, Inverse system of a symbolic power. III. Thin algebras and fat points, Compositio Math. 108 (1997), no. 3, 319–356. MR 1473851, DOI 10.1023/A:1000155612073
I. Jafarloo and G. Zito, On the containment problem for fat points, arXiv:1802.10178 (2018).
Antonio Laface and Luca Ugaglia, On a class of special linear systems of $\Bbb P^3$, Trans. Amer. Math. Soc. 358 (2006), no. 12, 5485–5500. MR 2238923, DOI 10.1090/S0002-9947-06-03891-8
Grzegorz Malara, Tomasz Szemberg, and Justyna Szpond, On a conjecture of Demailly and new bounds on Waldschmidt constants in $\Bbb {P}^N$, J. Number Theory 189 (2018), 211–219. MR 3788648, DOI 10.1016/j.jnt.2017.12.004
Rosa M. Miró-Roig, Harbourne, Schenck and Seceleanu’s conjecture, J. Algebra 462 (2016), 54–66. MR 3519499, DOI 10.1016/j.jalgebra.2016.05.020
Juan C. Migliore and Rosa María Miró-Roig, On the strong Lefschetz problem for uniform powers of general linear forms in $k[x,y,z]$, Proc. Amer. Math. Soc. 146 (2018), no. 2, 507–523. MR 3731687, DOI 10.1090/proc/13747
Juan C. Migliore, Rosa M. Miró-Roig, and Uwe Nagel, Monomial ideals, almost complete intersections and the weak Lefschetz property, Trans. Amer. Math. Soc. 363 (2011), no. 1, 229–257. MR 2719680, DOI 10.1090/S0002-9947-2010-05127-X
Juan C. Migliore, Rosa M. Miró-Roig, and Uwe Nagel, On the weak Lefschetz property for powers of linear forms, Algebra Number Theory 6 (2012), no. 3, 487–526. MR 2966707, DOI 10.2140/ant.2012.6.487
J. Migliore and U. Nagel, The Lefschetz question for ideals generated by powers of linear forms in few variables, arXiv:1703.07456 (2017). J. Commut. Algebra (to appear).
Masayoshi Nagata, On the fourteenth problem of Hilbert, Proc. Internat. Congress Math. 1958., Cambridge Univ. Press, New York, 1960, pp. 459–462. MR 0116056
Uwe Nagel, On the minimal free resolution of $r+3$ points in projective $r$-space, J. Pure Appl. Algebra 96 (1994), no. 1, 23–38. MR 1297438, DOI 10.1016/0022-4049(94)90084-1
Les Reid, Leslie G. Roberts, and Moshe Roitman, On complete intersections and their Hilbert functions, Canad. Math. Bull. 34 (1991), no. 4, 525–535. MR 1136655, DOI 10.4153/CMB-1991-083-9
Hal Schenck and Alexandra Seceleanu, The weak Lefschetz property and powers of linear forms in $\Bbb K[x,y,z]$, Proc. Amer. Math. Soc. 138 (2010), no. 7, 2335–2339. MR 2607862, DOI 10.1090/S0002-9939-10-10288-3
I. J. Schoenberg, Contributions to the problem of approximation of equidistant data by analytic functions. Part B. On the problem of osculatory interpolation. A second class of analytic approximation formulae, Quart. Appl. Math. 4 (1946), 112–141. MR 16705, DOI 10.1090/S0033-569X-1946-16705-2
I. J. Schoenberg, Cardinal spline interpolation, Conference Board of the Mathematical Sciences Regional Conference Series in Applied Mathematics, No. 12, Society for Industrial and Applied Mathematics, Philadelphia, Pa., 1973. MR 0420078, DOI 10.1137/1.9781611970555
Richard P. Stanley, Weyl groups, the hard Lefschetz theorem, and the Sperner property, SIAM J. Algebraic Discrete Methods 1 (1980), no. 2, 168–184. MR 578321, DOI 10.1137/0601021
Richard P. Stanley, The number of faces of a simplicial convex polytope, Adv. in Math. 35 (1980), no. 3, 236–238. MR 563925, DOI 10.1016/0001-8708(80)90050-X
Bernd Sturmfels and Zhiqiang Xu, Sagbi bases of Cox-Nagata rings, J. Eur. Math. Soc. (JEMS) 12 (2010), no. 2, 429–459. MR 2608947, DOI 10.4171/JEMS/204
Michel Waldschmidt, Propriétés arithmétiques de fonctions de plusieurs variables. II, Séminaire Pierre Lelong (Analyse) année 1975/76, Lecture Notes in Math., Vol. 578, Springer, Berlin, 1977, pp. 108–135 (French). MR 0453659
Ren-Hong Wang, Yan Xu, and Zhi-Qiang Xu, Eulerian numbers: a spline perspective, J. Math. Anal. Appl. 370 (2010), no. 2, 486–490. MR 2651669, DOI 10.1016/j.jmaa.2010.05.017
Junzo Watanabe, The Dilworth number of Artinian rings and finite posets with rank function, Commutative algebra and combinatorics (Kyoto, 1985) Adv. Stud. Pure Math., vol. 11, North-Holland, Amsterdam, 1987, pp. 303–312. MR 951211, DOI 10.2969/aspm/01110303
Yan Xu and Ren-hong Wang, Asymptotic properties of $B$-splines, Eulerian numbers and cube slicing, J. Comput. Appl. Math. 236 (2011), no. 5, 988–995. MR 2853521, DOI 10.1016/j.cam.2011.08.003