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Unexpected curves in \(\mathbb{P}^2\), line arrangements, and minimal degree of Jacobian relations. (English) Zbl 1525.14067

In the paper under review, the author delivers results devoted to the existence of unexpected curves via the minimal degree of non-trivial of Jacobian relations.
Let \(Z = \{P_{1},\dots, P_{d}\}\subset \mathbb{P}^{2}_{\mathbb{C}}\) be a finite set of \(d\) points. We say that \(Z\) admits unexpected curves of degree \(j\geq 2\) if \[ h^{0}(\mathbb{P}^{2}_{\mathbb{C}}, \mathcal{O}_{\mathbb{P}^{2}_{\mathbb{C}}}(j) \otimes \mathcal{I}(Z + (j-1)q)) > \max\bigg(0, h^{0}(\mathbb{P}^{2}_{\mathbb{C}},\mathcal{O}_{\mathbb{P}^{2}_{\mathbb{C}}}(j) \otimes \mathcal{I}(Z) - \binom{j}{2}\bigg), \] where \(q\) is a generic point and the fat point scheme \(kq\) is defined by the \(k\)-th power of the corresponding maximal ideal sheaf \(\mathcal{I}(q)\). Let \(\mathcal{A}_{Z} \, : \, f_{Z}=0\) be the associated line arrangement in the dual projective plane and let \((a_{Z}, b_{Z})\) be the generic splitting type of the derivation bundle \(E_{Z}\) associated to \(\mathcal{A}_{Z}\). Denote by \(m(\mathcal{A}_{Z})\) the maximal multiplicity of an intersection point in \(\mathcal{A}_{Z}\). Recall that the minimal degree of a Jacobian syzygy for the polynomial \(f\) is the integer \(\mathrm{mdr}(f)\) defined to be the smallest integer \(r\geq 0\) such that there exists a non-trivial relation \[ a\partial_{x} \, f + b\partial_{y} \, f + c\partial_{z} \, f = 0 \] among the partial derivatives with coefficients in \(a,b,c \in \mathbb{C}[x,y,z]_{r}\).
The main result of this paper under review can be formulated as follows.
Main Theorem. The set of points \(Z\) admits an unexpected curve if and only if \[ m(\mathcal{A}_{Z}) \leq\mathrm{mdr}(f_{Z}) + 1 < \frac{d}{2}. \] If these conditions are fulfilled, then \(Z\) admits an unexpected curve of degree \(j\) if and only if \[ \mathrm{mdr}(f_{Z}) < j \leq d - \mathrm{mdr}(f_{Z})-2. \] Using this result, the author presents some interesting applications. For example, the author shows that the irreducible unexpected quintics can only occur when the set of points \(Z\) has cardinality equal to \(11\) or \(12\).

MSC:

14N20 Configurations and arrangements of linear subspaces
13D02 Syzygies, resolutions, complexes and commutative rings
32S22 Relations with arrangements of hyperplanes
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References:

[1] T. Abe, “Restrictions of free arrangements and the division theorem”, pp. 389-401 in Perspectives in Lie theory, edited by F. Callegaro et al., Springer INdAM Ser. 19, Springer, 2017. · Zbl 1391.32044 · doi:10.1007/978-3-319-58971-8_14
[2] T. Abe, “Double points of free projective line arrangements”, Int. Math. Res. Not. 2022:3 (2022), 1811-1824. · Zbl 1483.14096 · doi:10.1093/imrn/rnaa145
[3] T. Abe and A. Dimca, “Splitting types of bundles of logarithmic vector fields along plane curves”, Internat. J. Math. 29:8 (2018), art. id. 1850055. · Zbl 1394.14020 · doi:10.1142/S0129167X18500556
[4] T. Abe and A. Dimca, “On complex supersolvable line arrangements”, J. Algebra 552 (2020), 38-51. · Zbl 1436.14053 · doi:10.1016/j.jalgebra.2020.02.007
[5] T. Abe, A. Dimca, and G. Sticlaru, “Addition-deletion results for the minimal degree of logarithmic derivations of hyperplane arrangements and maximal Tjurina line arrangements”, J. Algebraic Combin. 54:3 (2021), 739-766. · Zbl 1482.14056 · doi:10.1007/s10801-020-00986-9
[6] S. Akesseh, Ideal containments under flat extensions and interpolation on linear systems in \[ \mathbb{P}^2\], Ph.D. thesis, University of Nebraska-Lincoln, 2017, available at https://www.proquest.com/docview/1947103023. · Zbl 1408.13053
[7] B. Anzis and Ş. O. Tohǎneanu, “On the geometry of real or complex supersolvable line arrangements”, J. Combin. Theory Ser. A 140 (2016), 76-96. · Zbl 1335.52031 · doi:10.1016/j.jcta.2016.01.001
[8] M. Barakat, R. Behrends, C. Jefferson, L. Kühne, and M. Leuner, “On the generation of rank \[3\] simple matroids with an application to Terao’s freeness conjecture”, SIAM J. Discrete Math. 35:2 (2021), 1201-1223. · Zbl 1465.05025 · doi:10.1137/19M1296744
[9] T. Bauer, G. Malara, T. Szemberg, and J. Szpond, “Quartic unexpected curves and surfaces”, Manuscripta Math. 161:3-4 (2020), 283-292. · Zbl 1432.14031 · doi:10.1007/s00229-018-1091-3
[10] D. Cook, II, B. Harbourne, J. Migliore, and U. Nagel, “Line arrangements and configurations of points with an unexpected geometric property”, Compos. Math. 154:10 (2018), 2150-2194. · Zbl 1408.14174 · doi:10.1112/s0010437x18007376
[11] R. Di Gennaro, G. Ilardi, and J. Vallès, “Singular hypersurfaces characterizing the Lefschetz properties”, J. Lond. Math. Soc. (2) 89:1 (2014), 194-212. · Zbl 1290.13013 · doi:10.1112/jlms/jdt053
[12] M. Di Marca, G. Malara, and A. Oneto, “Unexpected curves arising from special line arrangements”, J. Algebraic Combin. 51:2 (2020), 171-194. · Zbl 1433.14049 · doi:10.1007/s10801-019-00871-0
[13] A. Dimca, Singularities and topology of hypersurfaces, Springer, 1992. · Zbl 0753.57001 · doi:10.1007/978-1-4612-4404-2
[14] A. Dimca, “Curve arrangements, pencils, and Jacobian syzygies”, Michigan Math. J. 66:2 (2017), 347-365. · Zbl 1375.14106 · doi:10.1307/mmj/1490639821
[15] A. Dimca, “Freeness versus maximal global Tjurina number for plane curves”, Math. Proc. Cambridge Philos. Soc. 163:1 (2017), 161-172. · Zbl 1387.14080 · doi:10.1017/S0305004116000803
[16] A. Dimca, Hyperplane arrangements: an introduction, Springer, 2017. · Zbl 1362.14001 · doi:10.1007/978-3-319-56221-6
[17] A. Dimca, “On the minimal value of global Tjurina numbers for line arrangements”, Eur. J. Math. 6:3 (2020), 817-828. · Zbl 1446.14018 · doi:10.1007/s40879-019-00373-0
[18] A. Dimca and M. Saito, “Some remarks on limit mixed Hodge structures and spectrum”, An. Ştiinţ. Univ. “Ovidius” Constanţa Ser. Mat. 22:2 (2014), 69-78. · Zbl 1389.14005 · doi:10.2478/auom-2014-0032
[19] A. Dimca and M. Saito, “Generalization of theorems of Griffiths and Steenbrink to hypersurfaces with ordinary double points”, Bull. Math. Soc. Sci. Math. Roumanie (N.S.) 60(108):4 (2017), 351-371. · Zbl 1399.32019
[20] A. Dimca and E. Sernesi, “Syzygies and logarithmic vector fields along plane curves”, J. Éc. polytech. Math. 1 (2014), 247-267. · Zbl 1327.14049 · doi:10.5802/jep.10
[21] A. Dimca and G. Sticlaru, “Free and nearly free curves vs. rational cuspidal plane curves”, Publ. Res. Inst. Math. Sci. 54:1 (2018), 163-179. · Zbl 1391.14057 · doi:10.4171/PRIMS/54-1-6
[22] A. Dimca and G. Sticlaru, “On supersolvable and nearly supersolvable line arrangements”, J. Algebraic Combin. 50:4 (2019), 363-378. · Zbl 1437.14037 · doi:10.1007/s10801-018-0859-6
[23] A. Dimca and G. Sticlaru, “Plane curves with three syzygies, minimal Tjurina curves, and nearly cuspidal curves”, Geom. Dedicata 207 (2020), 29-49. · Zbl 1505.14073 · doi:10.1007/s10711-019-00485-7
[24] A. Dimca, D. Ibadula, and D. A. Măcinic, “Numerical invariants and moduli spaces for line arrangements”, Osaka J. Math. 57:4 (2020), 847-870. · Zbl 1457.32075
[25] Ł. Farnik, F. Galuppi, L. Sodomaco, and W. Trok, “On the unique unexpected quartic in \[\mathbb{P}^2\]”, J. Algebraic Combin. 53:1 (2021), 131-146. · Zbl 1461.14073 · doi:10.1007/s10801-019-00922-6
[26] K. Hanumanthu and B. Harbourne, “Real and complex supersolvable line arrangements in the projective plane”, J. Algebraic Combin. 54:3 (2021), 767-785. · Zbl 1475.14107 · doi:10.1007/s10801-020-00987-8
[27] B. Harbourne, J. Migliore, U. Nagel, and Z. Teitler, “Unexpected hypersurfaces and where to find them”, Michigan Math. J. 70:2 (2021), 301-339. · Zbl 1469.14107 · doi:10.1307/mmj/1593741748
[28] F. Hirzebruch, “Arrangements of lines and algebraic surfaces”, pp. 113-140 in Arithmetic and geometry, II: Geometry, edited by M. Artin and J. Tate, Progr. Math. 36, Birkhäuser, Boston, MA, 1983. · Zbl 0527.14033 · doi:10.1007/978-1-4757-9286-7_7
[29] J. Kollár, “Singularities of pairs”, pp. 221-287 in Algebraic geometry (Santa Cruz, CA, 1995), edited by J. Kollár et al., Proc. Sympos. Pure Math. 62, Amer. Math. Soc., Providence, RI, 1997. · Zbl 0905.14002 · doi:10.1090/pspum/062.1/1492525
[30] Y. Matsumoto, S. Moriyama, H. Imai, and D. Bremner, “Matroid enumeration for incidence geometry”, Discrete Comput. Geom. 47:1 (2012), 17-43. · Zbl 1236.05055 · doi:10.1007/s00454-011-9388-y
[31] A. A. du Plessis and C. T. C. Wall, “Application of the theory of the discriminant to highly singular plane curves”, Math. Proc. Cambridge Philos. Soc. 126:2 (1999), 259-266. · Zbl 0926.14012 · doi:10.1017/S0305004198003302
[32] J. Szpond, “Fermat-type arrangements”, pp. 161-182 in Combinatorial structures in algebra and geometry (Constanţa, Romania, 2018), edited by D. I. Stamate and T. Szemberg, Springer Proc. Math. Stat. 331, Springer, 2020. · Zbl 1453.14022 · doi:10.1007/978-3-030-52111-0_12
[33] Ş. O. Tohǎneanu, “A computational criterion for the supersolvability of line arrangements”, Ars Combin. 117 (2014), 217-223 · Zbl 1340.52027
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