Freeness and invariants of rational plane curves
HTML articles powered by AMS MathViewer
- by Laurent Busé, Alexandru Dimca and Gabriel Sticlaru HTML | PDF
- Math. Comp. 89 (2020), 1525-1546 Request permission
Abstract:
Given a parameterization $\phi$ of a rational plane curve $\mathcal {C}$, we study some invariants of $\mathcal {C}$ via $\phi$. We first focus on the characterization of rational cuspidal curves, in particular, we establish a relation between the discriminant of the pull-back of a line via $\phi$, the dual curve of $\mathcal {C}$, and its singular points. Then, by analyzing the pull-backs of the global differential forms via $\phi$, we prove that the (nearly) freeness of a rational curve can be tested by inspecting the Hilbert function of the kernel of a canonical map. As a by-product, we also show that the global Tjurina number of a rational curve can be computed directly from one of its parameterization, without relying on the computation of an equation of $\mathcal {C}$.References
- Enrique Artal Bartolo, Leire Gorrochategui, Ignacio Luengo, and Alejandro Melle-Hernández, On some conjectures about free and nearly free divisors, Singularities and computer algebra, Springer, Cham, 2017, pp. 1–19. MR 3675719
- Laurent Busé and Carlos D’Andrea, Singular factors of rational plane curves, J. Algebra 357 (2012), 322–346. MR 2905259, DOI 10.1016/j.jalgebra.2012.01.030
- Alessandra Bernardi, Alessandro Gimigliano, and Monica Idà, Singularities of plane rational curves via projections, J. Symbolic Comput. 86 (2018), 189–214. MR 3725220, DOI 10.1016/j.jsc.2017.05.003
- Laurent Busé and Jean-Pierre Jouanolou, On the closed image of a rational map and the implicitization problem, J. Algebra 265 (2003), no. 1, 312–357. MR 1984914, DOI 10.1016/S0021-8693(03)00181-9
- Laurent Busé, On the equations of the moving curve ideal of a rational algebraic plane curve, J. Algebra 321 (2009), no. 8, 2317–2344. MR 2501523, DOI 10.1016/j.jalgebra.2009.01.030
- David Cox, Andrew R. Kustin, Claudia Polini, and Bernd Ulrich, A study of singularities on rational curves via syzygies, Mem. Amer. Math. Soc. 222 (2013), no. 1045, x+116. MR 3059370, DOI 10.1090/S0065-9266-2012-00674-5
- David A. Cox, Thomas W. Sederberg, and Falai Chen, The moving line ideal basis of planar rational curves, Comput. Aided Geom. Design 15 (1998), no. 8, 803–827. MR 1638732, DOI 10.1016/S0167-8396(98)00014-4
- Alexandru Dimca and Gert-Martin Greuel, On 1-forms on isolated complete intersection curve singularities, J. Singul. 18 (2018), 114–118. MR 3899537, DOI 10.1007/bf01405360
- Alexandru Dimca, Singularities and topology of hypersurfaces, Universitext, Springer-Verlag, New York, 1992. MR 1194180, DOI 10.1007/978-1-4612-4404-2
- Alexandru Dimca, On polar Cremona transformations, An. Ştiinţ. Univ. Ovidius Constanţa Ser. Mat. 9 (2001), no. 1, 47–53. To Mirela Ştefănescu, at her 60’s. MR 1946153
- A. Dimca, D. Ibadula, and A. Macinic, Numerical invariants and moduli spaces for line arrangements. Preprint arXiv:1609.06551, to appear in Osaka Math. J., 2016.
- Alexandru Dimca, Freeness versus maximal global Tjurina number for plane curves, Math. Proc. Cambridge Philos. Soc. 163 (2017), no. 1, 161–172. MR 3656354, DOI 10.1017/S0305004116000803
- A. Dimca, On rational cuspidal plane curves, and the local cohomology of jacobian rings. Preprint arXiv:1707.05258, to appear in Comm. Math. Helv., 2017.
- Alexandru Dimca and Stefan Papadima, Hypersurface complements, Milnor fibers and higher homotopy groups of arrangments, Ann. of Math. (2) 158 (2003), no. 2, 473–507. MR 2018927, DOI 10.4007/annals.2003.158.473
- 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 (1999), no. 2, 259–266. MR 1670229, DOI 10.1017/S0305004198003302
- Alexandru Dimca and Edoardo Sernesi, Syzygies and logarithmic vector fields along plane curves, J. Éc. polytech. Math. 1 (2014), 247–267 (English, with English and French summaries). MR 3322789, DOI 10.5802/jep.10
- Alexandru Dimca and Gabriel Sticlaru, Free divisors and rational cuspidal plane curves, Math. Res. Lett. 24 (2017), no. 4, 1023–1042. MR 3723802, DOI 10.4310/MRL.2017.v24.n4.a5
- Alexandru Dimca and Gabriel Sticlaru, On the exponents of free and nearly free projective plane curves, Rev. Mat. Complut. 30 (2017), no. 2, 259–268. MR 3642034, DOI 10.1007/s13163-017-0228-3
- Alexandru Dimca and Gabriel Sticlaru, Free and nearly free curves vs. rational cuspidal plane curves, Publ. Res. Inst. Math. Sci. 54 (2018), no. 1, 163–179. MR 3749348, DOI 10.4171/PRIMS/54-1-6
- Alexandru Dimca and Gabriel Sticlaru, On the freeness of rational cuspidal plane curves, Mosc. Math. J. 18 (2018), no. 4, 659–666. MR 3914108, DOI 10.17323/1609-4514-2018-18-4-659-666
- David Eisenbud, Commutative algebra, Graduate Texts in Mathematics, vol. 150, Springer-Verlag, New York, 1995. With a view toward algebraic geometry. MR 1322960, DOI 10.1007/978-1-4612-5350-1
- Thiago Fassarella and Nivaldo Medeiros, On the polar degree of projective hypersurfaces, J. Lond. Math. Soc. (2) 86 (2012), no. 1, 259–271. MR 2959304, DOI 10.1112/jlms/jds005
- Hajime Kaji, The separability of the Gauss map versus the reflexivity, Geom. Dedicata 139 (2009), 75–82. MR 2481838, DOI 10.1007/s10711-008-9334-1
- Steven L. Kleiman, The enumerative theory of singularities, Real and complex singularities (Proc. Ninth Nordic Summer School/NAVF Sympos. Math., Oslo, 1976) Sijthoff and Noordhoff, Alphen aan den Rijn, 1977, pp. 297–396. MR 0568897
- T. K. Moe, Rational cuspidal curves. arXiv:1511.02691 (139 pages, Master thesis, University of Oslo), 2008.
- S. Marchesi and J. Vallès, Nearly free curves and arrangements: a vector bundle point of view, Math. Proc. Camb. Phil. Soc., 2019.
- Ramakrishna Nanduri, A family of irreducible free divisors in $\Bbb {P}^2$, J. Algebra Appl. 14 (2015), no. 7, 1550105, 11. MR 3339404, DOI 10.1142/S0219498815501054
- Aron Simis and Ştefan O. Tohǎneanu, Homology of homogeneous divisors, Israel J. Math. 200 (2014), no. 1, 449–487. MR 3219587, DOI 10.1007/s11856-014-0025-3
Additional Information
- Laurent Busé
- Affiliation: Université Côte d’Azur; and Inria, Sophia Antipolis, France
- Email: laurent.buse@inria.fr
- Alexandru Dimca
- Affiliation: Université Côte d’Azur, Laboratoire Jean-Alexandre Dieudonné; and Inria, Nice, France
- MR Author ID: 58125
- Email: alexandru.dimca@unice.fr
- Gabriel Sticlaru
- Affiliation: Faculty of Mathematics and Informatics, Ovidius University, Bd. Mamaia 124, 900527 Constanta, Romania
- MR Author ID: 997310
- Email: gabrielsticlaru@yahoo.com
- Received by editor(s): April 26, 2018
- Received by editor(s) in revised form: February 26, 2019
- Published electronically: December 16, 2019
- Additional Notes: This work was partially supported by the French government through the $\mathrm {UCA}^\mathrm {JEDI}$ Investments in the Future project managed by the National Research Agency (ANR) with the reference number ANR-15-IDEX-01
- © Copyright 2019 American Mathematical Society
- Journal: Math. Comp. 89 (2020), 1525-1546
- MSC (2010): Primary 14H50; Secondary 14H20, 14H45
- DOI: https://doi.org/10.1090/mcom/3495
- MathSciNet review: 4063327