×

On the containment problem for fat points ideals and Harbourne’s conjecture. (English) Zbl 1442.13052

In the paper under review, the authors study Harbourne’s conjecture on the containment of symbolic powers of homogeneous ideals. Let \(R = \mathbb{K}[x_{0},\dots, x_{n}]\) be a polynomial ring over a field \(\mathbb{K}\) and let \(I \subset R\) be a homogeneous ideal. We define the \(m\)-th symbolic power of \(I\) as \[ I^{(m)} = R \cap \bigg( \bigcap_{P \in\mathrm{Ass}(I)} (I^{m})_{P}\bigg). \] If we restrict our setting to fat points, i.e., \[ I = \bigcap_{i=1}^{s} I(P_{i})^{m_{i}}, \] where \(\mathcal{P} = \{P_{1},\dots, P_{s}\}\) is a set of distinct points in \(\mathbb{P}^{N}_{\mathbb{K}}\), then \[ I^{(m)} = \bigcap_{i=1}^{s} I(P_{i})^{m \cdot m_{i}}. \] For a homogeneous ideal \(I\) we denote by \(\alpha(I)\) the initial degree of \(I\).
The crucial spot in the paper is played by the following conjecture by Harbourne.
Conjecture. Let \(I\) be a homogeneous ideal in \(R\). Then for all \(m>0\) one has \[ I^{(Nm-N+1)} \subseteq I^{m}. \]
The first main result of the paper provides a positive answer for Harbourne’s conjecture when \(\mathcal{P}\) is a finite set of very general points in \(\mathbb{P}^{N}_{\mathbb{K}}\) and for sufficiently large values \(m>0\).
Theorem A. Let \(\mathcal{P} = \{P_{1},\dots, P_{s}\}\) be a finite set of very general points and \(I = \bigcap_{i=1}^{s}I(P_{i})\) the associated ideal. Set \[ \beta= \frac{(N-1)(\alpha(I)+N-1)}{(N-2)N} \] if \(N\geq 4\) or \(\beta = 1\) provided that \(N=2,3\). Then for \(m\geq \beta\) one has \[ I^{(Nm-N+1)}\subseteq I^{m}. \] The second main result of the paper is devoted to fat point schemes constructed via singular loci of line arrangements in the plane. Let \(\mathcal{A} \subset \mathbb{P}^{2}_{\mathbb{K}}\) be an arrangement of \(n\) lines and we fix the linear forms \(\ell_{1}, \dots, \ell_{n} \in R\) defining the lines. Suppose that \(\mathrm{ht}(\langle \ell_{1},\dots, \ell_{n}\rangle) = 3\), and we define the ideal \[ I = \langle \ell_{2} \cdots \ell_{n}, \ell_{1}\ell_{3} \cdots \ell_{n},\dots, \ell_{1}\ell_{2} \cdots \ell_{n-1} \rangle. \] It is worth noticing that \(I\) has the primary decomposition \[ I = I(P_{1})^{n_{1}-1} \cap\dots\cap I(P_{s})^{n_{s}-1}, \] where \(P_{1},\dots, P_{s}\) is the singular locus of the arrangement \(\mathcal{A}\) and \(n_{i}\) denotes the multiplicity of \(P_{i}\), i.e., the number of lines from \(\mathcal{A}\) passing through \(P_{i}\). Finally, we denote by \(\mathfrak{m} = \langle x,y,z\rangle\).
Theorem B. In the setting as above, for every \(r\geq 1\) one has
i) \(I^{(2r-1)}\subseteq \mathfrak{m}^{r-1} \cdot I^{r}\), and
ii) \(I^{(2r)} \subseteq \mathfrak{m}^{r} \cdot I^{r}\).

MSC:

13F20 Polynomial rings and ideals; rings of integer-valued polynomials
13A02 Graded rings
13A15 Ideals and multiplicative ideal theory in commutative rings
14N07 Secant varieties, tensor rank, varieties of sums of powers
14N20 Configurations and arrangements of linear subspaces

Software:

Macaulay2
PDFBibTeX XMLCite
Full Text: DOI arXiv

References:

[1] Bauer, Thomas; Di Rocco, Sandra; Harbourne, Brian; Kapustka, Micha\l; Knutsen, Andreas; Syzdek, Wioletta; Szemberg, Tomasz, A primer on Seshadri constants. Interactions of classical and numerical algebraic geometry, Contemp. Math. 496, 33-70 (2009), Amer. Math. Soc., Providence, RI · Zbl 1184.14008 · doi:10.1090/conm/496/09718
[2] Bocci, Cristiano; Cooper, Susan M.; Harbourne, Brian, Containment results for ideals of various configurations of points in \(\mathbf{P}^N\), J. Pure Appl. Algebra, 218, 1, 65-75 (2014) · Zbl 1285.13029 · doi:10.1016/j.jpaa.2013.04.012
[3] Bocci, Cristiano; Harbourne, Brian, Comparing powers and symbolic powers of ideals, J. Algebraic Geom., 19, 3, 399-417 (2010) · Zbl 1198.14001 · doi:10.1090/S1056-3911-09-00530-X
[4] Bocci, Cristiano; Harbourne, Brian, The resurgence of ideals of points and the containment problem, Proc. Amer. Math. Soc., 138, 4, 1175-1190 (2010) · Zbl 1200.14018 · doi:10.1090/S0002-9939-09-10108-9
[5] Chandler, Karen A., Regularity of the powers of an ideal, Comm. Algebra, 25, 12, 3773-3776 (1997) · Zbl 0928.14033 · doi:10.1080/00927879708826084
[6] Derksen, Harm; Sidman, Jessica, A sharp bound for the Castelnuovo-Mumford regularity of subspace arrangements, Adv. Math., 172, 2, 151-157 (2002) · Zbl 1040.13009 · doi:10.1016/S0001-8708(02)00019-1
[7] Dumnicki, Marcin, Containments of symbolic powers of ideals of generic points in \(\mathbb{P}^3\), Proc. Amer. Math. Soc., 143, 2, 513-530 (2015) · Zbl 1342.14122 · doi:10.1090/S0002-9939-2014-12273-8
[8] Dumnicki, Marcin; Szemberg, Tomasz; Tutaj-Gasi\'{n}ska, Halszka, Counterexamples to the \(I^{(3)}\subset I^2\) containment, J. Algebra, 393, 24-29 (2013) · Zbl 1297.14008 · doi:10.1016/j.jalgebra.2013.06.039
[9] Dumnicki, M.; Szemberg, T.; Tutaj-Gasi\'{n}ska, H., A vanishing theorem and symbolic powers of planar point ideals, LMS J. Comput. Math., 16, 373-387 (2013) · Zbl 1319.14008 · doi:10.1112/S1461157013000181
[10] Dumnicki, Marcin; Tutaj-Gasi\'{n}ska, Halszka, A containment result in \(P^n\) and the Chudnovsky conjecture, Proc. Amer. Math. Soc., 145, 9, 3689-3694 (2017) · Zbl 1374.13005 · doi:10.1090/proc/13582
[11] Ein, Lawrence; Lazarsfeld, Robert; Smith, Karen E., Uniform bounds and symbolic powers on smooth varieties, Invent. Math., 144, 2, 241-252 (2001) · Zbl 1076.13501 · doi:10.1007/s002220100121
[12] Fouli, Louiza; Mantero, Paolo; Xie, Yu, Chudnovsky’s conjecture for very general points in \(\mathbb{P}^N_k\), J. Algebra, 498, 211-227 (2018) · Zbl 1386.13058 · doi:10.1016/j.jalgebra.2017.11.002
[13] Geramita, A. V.; Harbourne, B.; Migliore, J., Star configurations in \(\mathbb{P}^n\), J. Algebra, 376, 279-299 (2013) · Zbl 1284.14073 · doi:10.1016/j.jalgebra.2012.11.034
[14] GrSt D. Grayson and M. Stillman, Macaulay2, a software system for research in algebraic geometry, Available at http://www.math.uiuc.edu/Macaulay2/.
[15] G18 E. Grifo, A stable version of Harbourne’s Conjecture and the containment problem for space monomial curves, arXiv:1809.06955. · Zbl 1453.13009
[16] Grifo, Elo\'{\i}sa; Huneke, Craig, Symbolic powers of ideals defining F-pure and strongly F-regular rings, Int. Math. Res. Not. IMRN, 10, 2999-3014 (2019) · Zbl 1505.13008 · doi:10.1093/imrn/rnx213
[17] Harbourne, Brian; Huneke, Craig, Are symbolic powers highly evolved?, J. Ramanujan Math. Soc., 28A, 247-266 (2013) · Zbl 1296.13018
[18] Hochster, Melvin; Huneke, Craig, Comparison of symbolic and ordinary powers of ideals, Invent. Math., 147, 2, 349-369 (2002) · Zbl 1061.13005 · doi:10.1007/s002220100176
[19] JaZi I. B. Jafarloo and G. Zito, On the containment problem for fat points, J. Commut. Algebra (2019), to appear. · Zbl 1487.14118
[20] Schenck, Hal, Resonance varieties via blowups of \(\mathbb{P}^2\) and scrolls, Int. Math. Res. Not. IMRN, 20, 4756-4778 (2011) · Zbl 1230.14081
[21] Szemberg, T.; Szpond, J., On the containment problem, Rend. Circ. Mat. Palermo (2), 66, 2, 233-245 (2017) · Zbl 1386.14045 · doi:10.1007/s12215-016-0281-7
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.