On plane rational curves and the splitting of the tangent bundle
Annali della Scuola Normale Superiore di Pisa - Classe di Scienze, Serie 5, Volume 12 (2013) no. 3, p. 587-621

Given an immersion $\varphi :{\mathbf{P}}^{1}\to {\mathbf{P}}^{2}$, we give new approaches to determining the splitting of the pullback of the cotangent bundle. We also give new bounds on the splitting type for immersions which factor as $\varphi :{\mathbf{P}}^{1}\cong D\subset X\to {\mathbf{P}}^{2}$, where $X\to {\mathbf{P}}^{2}$ is obtained by blowing up $r$ distinct points ${p}_{i}\in {\mathbf{P}}^{2}$. As applications in the case that the points ${p}_{i}$ are generic, we give a complete determination of the splitting types for such immersions when $r\le 7$. The case that ${D}^{2}=-1$ is of particular interest. For $r\le 8$ generic points, it is known that there are only finitely many inequivalent $\varphi$ with ${D}^{2}=-1$, and all of them have balanced splitting. However, for $r=9$ generic points we show that there are infinitely many inequivalent $\varphi$ with ${D}^{2}=-1$ having unbalanced splitting (only two such examples were known previously). We show that these new examples are related to a semi-adjoint formula which we conjecture accounts for all occurrences of unbalanced splitting when ${D}^{2}=-1$ in the case of $r=9$ generic points ${p}_{i}$. In the last section we apply such results to the study of the resolution of fat point schemes.

Published online : 2019-02-21
Classification:  14C20,  13P10,  14J26,  14J60
@article{ASNSP_2013_5_12_3_587_0,
author = {Gimigliano, Alessandro and Harbourne, Brian and Id\a, Monica},
title = {On plane rational curves and the splitting of the tangent bundle},
journal = {Annali della Scuola Normale Superiore di Pisa - Classe di Scienze},
publisher = {Scuola Normale Superiore, Pisa},
volume = {Ser. 5, 12},
number = {3},
year = {2013},
pages = {587-621},
zbl = {06232457},
mrnumber = {3137457},
language = {en},
url = {http://www.numdam.org/item/ASNSP_2013_5_12_3_587_0}
}

Gimigliano, Alessandro; Harbourne, Brian; Idà, Monica. On plane rational curves and the splitting of the tangent bundle. Annali della Scuola Normale Superiore di Pisa - Classe di Scienze, Serie 5, Volume 12 (2013) no. 3, pp. 587-621. http://www.numdam.org/item/ASNSP_2013_5_12_3_587_0/`

[1] M.-G. Ascenzi, The restricted tangent bundle of a rational curve in ${\mathbf{P}}^{2}$, Comm. Algebra 16 (1988), 2193–2208. | MR 962311 | Zbl 0675.14010

[2] M.-G. Ascenzi, The restricted tangent bundle of a rational curve on a quadric in ${P}^{3}$, Proc. Amer. Math. Soc. 98 (1986), 561–566. | MR 861750 | Zbl 0616.14027

[3] G. Birkhoff, A theorem on matrices of analytic functions, Math. Ann. 74 (1913), 122–133. | JFM 44.0469.02 | MR 1511753

[4] H. Clemens, On rational curves in $n$-space with given normal bundle, In: “Advances in Algebraic Geometry Motivated by Physics” (Lowell, MA, 2000), Contemp. Math., 276, Amer. Math. Soc., Providence, RI, 2001, 137–144. | MR 1837114 | Zbl 0993.14011

[5] D. Cox, T. W. Sederburg and F. Chen, The moving line ideal basis of planar rational curves, Comput. Aided Geom. Design 15 (1998), 803–827. | MR 1638732 | Zbl 0908.68174

[6] T. de Fernex, Negative curves on very general blow-ups of ${\mathbf{P}}^{2}$, In: “Projective Varieties with Unexpected Properties”, M. Beltrametti et al. (eds.), a volume in memory of Giuseppe Veronese, de Gruyter, Berlin, 2005, 199–207. | MR 2202253 | Zbl 1121.14006

[7] T. de Fernex, On the Mori cone of blow-ups of the plane, preprint (arXiv:1001.5243).

[8] D. Eisenbud and A. Van de Ven, On the normal bundles of smooth rational space curves, Math. Ann. 256 (1981), 453–463. | MR 628227 | Zbl 0443.14015

[9] D. Eisenbud and A. Van de Ven, On the variety of smooth rational space curves with given degree and normal bundle, Invent. Math. 67 (1982), 89–100. | MR 664325 | Zbl 0492.14016

[10] S. Fitchett, On bounding the number of generators for fat point ideals on the projective plane, J. Algebra 236 (2001), 502–521. | MR 1813489 | Zbl 1007.14008

[11] S. Fitchett, Corrigendum to: On bounding the number of generators for fat point ideals on the projective plane [J. Algebra 236 (2001), 502–521], J. Algebra 276 (2004), 417–419. | MR 2054405 | Zbl 1007.14008

[12] S. Fitchett, B. Harbourne and S. Holay, Resolutions of fat point ideals involving eight general points of ${\mathbf{P}}^{2}$, J. Algebra 244 (2001), 684–705. | MR 1859044 | Zbl 1033.14031

[13] F. Ghione and G. Sacchiero, Normal bundles of rational curves in ${\mathbf{P}}^{3}$, Manuscripta Math. 33 (1980), 111–128. | MR 597814 | Zbl 0496.14021

[14] A. Gimigliano, “On Linear Systems of Plane Curves”, Thesis, Queen’s University, Kingston, 1987. | MR 2635606

[15] A. Gimigliano, B. Harbourne and M. Idà, Betti numbers for fat point ideals in the plane: a geometric approach, Trans. Amer. Math. Soc. 361 (2009), 1103–1127. | MR 2452836 | Zbl 1170.14006

[16] A. Gimigliano, B. Harbourne and M. Idà, The role of the cotangent bundle in resolving ideals of fat points in the plane, J. Pure Appl. Algebra 213 (2009), 203–214. | MR 2467397 | Zbl 1161.14012

[17] A. Gimigliano, B. Harbourne and M. Idà, Stable postulation and stable ideal generation: conjectures for fat points in the plane, Bull. Belg. Math . Soc. Simon Stevin 16 (2009), 853–860. | MR 2574365 | Zbl 1187.14010

[18] A. Grothendieck, Sur la classification des fibrés holomorphes sur la sphère de Riemann, Amer. J. Math. 79 (1957), 121–138. | MR 87176 | Zbl 0079.17001

[19] L. Gruson, R. Lazarsfeld and Ch. Peskine, On a theorem of Castelnuovo and the equations defining space curves, Invent. Math. 72 (1983), 491–506. | MR 704401 | Zbl 0565.14014

[20] B. Harbourne, Complete linear systems on rational surfaces, Trans. Amer. Math. Soc. 289 (1985), 213–226. | MR 779061 | Zbl 0609.14004

[21] B. Harbourne, An Algorithm for fat points on ${\mathbf{P}}^{2}$, Canad. J. Math. 52 (2000), 123– 140. | MR 1745704 | Zbl 0963.14021

[22] B. Harbourne, Global aspects of the geometry of surfaces, Ann. Univ. Paedagog. Crac. Stud. Math. 9 (2010), 5–41. | MR 2608654 | Zbl 1247.14006

[23] B. Harbourne, Blowings-up of ${\mathbf{P}}^{2}$ and their blowings-down, Duke Math. J. 52 (1985), 129–148. | MR 791295 | Zbl 0577.14025

[24] B. Harbourne, Very ample divisors on rational surfaces, Math. Ann. 272 (1985), 139–153. | MR 794097 | Zbl 0545.14003

[25] A. Hirschowitz, Une conjecture pour la cohomologie des diviseurs sur les surfaces rationelles génériques, J. Reine Angew. Math. 397 (1989), 208–213. | MR 993223 | Zbl 0686.14013

[26] K. Hulek, The normal bundle of a curve on a quadric, Math. Ann. 258 (1981), 201– 206. | MR 641825 | Zbl 0458.14011

[27] G. Ilardi, P. Supino and J. Valles, Geometry of syzygies via Poncelet varieties, Boll. Unione Mat. Ital. (9) 2 (2009), 579–589. | MR 2569292 | Zbl 1197.13013

[28] V. Kac, “Infinite Dimensional Lie Algebras”, Cambridge University Press, New York, 1994. | MR 1104219 | Zbl 0716.17022

[29] M. Lahyane and B. Harbourne, Irreducibility of ($-1$)-classes of anticanonical rational surfaces, Pac. J. Math. 218 (2005), 101–114. | MR 2224591 | Zbl 1109.14030

[30] Y. I. Manin, “Cubic Forms”, Mathematical Library 4, North-Holland, 1986. | MR 833513 | Zbl 0582.14010

[31] M. Nagata, On rational surfaces, II, Mem. Coll. Sci. Univ. Kyoto, Ser. A Math. 33 (1960), 271–293. | MR 126444 | Zbl 0100.16801

[32] Z. Ran, Normal bundles of rational curves in projective spaces, Asian J. Math. 11 (2007), 567–608. | MR 2402939 | Zbl 1163.14029

[33] L. Ramella, La stratification du schéma de Hilbert des courbes rationelles de ${\mathbf{P}}^{n}$ par le fibré tangent restreint, C.R. Acad. Sci. Paris Sér. I, Moth. 311 (1990), 181–184. | MR 1065888 | Zbl 0721.14014

[34] T. Sederburg, R. Goldman and H. Du, Implicitizing rational curves by the method of moving algebraic curves, J. Symb. Comput. 23 (1997), 153–175. | MR 1448692 | Zbl 0872.68193

[35] T. Sederberg, T. Saito, D. Qi and K. Klimaszewski, Curve implicitization using moving lines, Comput. Aided Geom. Design 11 (1994), 687-706. | MR 1305914 | Zbl 0814.65018

[36] B. Segre, Alcune questioni su insiemi finiti di punti in Geometria Algebrica, In: “Atti del Convegno Internaz. di Geom. Alg.”, Torino, 1961. | Zbl 0104.38903