Homogeneous interpolation on ten points
Authors:
Ciro Ciliberto and Rick Miranda
Journal:
J. Algebraic Geom. 20 (2011), 685-726
DOI:
https://doi.org/10.1090/S1056-3911-2011-00545-6
Published electronically:
February 8, 2011
MathSciNet review:
2819673
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Abstract |
References |
Additional Information
Abstract: In this paper we prove that for all pairs $(d,m)$ with $d/m\geq 174/55$, the linear system of plane curves of degree $d$ with ten general base points of multiplicity $m$ has the expected dimension.
References
- G. Castelnuovo: Ricerche generali sopra i sistemi lineari di curve piane, Mem. Accad. Sci. Torino, II 42 (1891).
- Ciro Ciliberto, Francesca Cioffi, Rick Miranda, and Ferruccio Orecchia, Bivariate Hermite interpolation and linear systems of plane curves with base fat points, Computer mathematics, Lecture Notes Ser. Comput., vol. 10, World Sci. Publ., River Edge, NJ, 2003, pp. 87–102. MR 2061827
- Ciro Ciliberto and Rick Miranda, Degenerations of planar linear systems, J. Reine Angew. Math. 501 (1998), 191–220. MR 1637857
- Ciro Ciliberto and Rick Miranda, Linear systems of plane curves with base points of equal multiplicity, Trans. Amer. Math. Soc. 352 (2000), no. 9, 4037–4050. MR 1637062, DOI https://doi.org/10.1090/S0002-9947-00-02416-8
- Ciro Ciliberto and Rick Miranda, Matching conditions for degenerating plane curves and applications, Projective varieties with unexpected properties, Walter de Gruyter, Berlin, 2005, pp. 177–197. MR 2202252
- C. Ciliberto and R. Miranda, The Segre and Harbourne-Hirschowitz conjectures, Applications of algebraic geometry to coding theory, physics and computation (Eilat, 2001) NATO Sci. Ser. II Math. Phys. Chem., vol. 36, Kluwer Acad. Publ., Dordrecht, 2001, pp. 37–51. MR 1866894
- Ciro Ciliberto and Rick Miranda, Nagata’s conjecture for a square or nearly-square number of points, Ric. Mat. 55 (2006), no. 1, 71–78. MR 2248163, DOI https://doi.org/10.1007/s11587-006-0005-y
- Tommaso de Fernex, Negative curves on very general blow-ups of $\Bbb P^2$, Projective varieties with unexpected properties, Walter de Gruyter, Berlin, 2005, pp. 199–207. MR 2202253
- M. Dumnicki: Reduction method for linear systems of plane curves with base fat points. arXiv:math.AG/0606716, 2006.
- Marcin Dumnicki and Witold Jarnicki, New effective bounds on the dimension of a linear system in $\Bbb P^2$, J. Symbolic Comput. 42 (2007), no. 6, 621–635. MR 2325918, DOI https://doi.org/10.1016/j.jsc.2007.01.004
- Laurent Evain, Computing limit linear series with infinitesimal methods, Ann. Inst. Fourier (Grenoble) 57 (2007), no. 6, 1947–1974 (English, with English and French summaries). MR 2377892
- Robert Friedman, Global smoothings of varieties with normal crossings, Ann. of Math. (2) 118 (1983), no. 1, 75–114. MR 707162, DOI https://doi.org/10.2307/2006955
- A. Gimigliano: On linear systems of plane curves, Thesis, Queen’s University, Kingston (1987).
- Brian Harbourne, The geometry of rational surfaces and Hilbert functions of points in the plane, Proceedings of the 1984 Vancouver conference in algebraic geometry, CMS Conf. Proc., vol. 6, Amer. Math. Soc., Providence, RI, 1986, pp. 95–111. MR 846019
- Brian Harbourne, Complete linear systems on rational surfaces, Trans. Amer. Math. Soc. 289 (1985), no. 1, 213–226. MR 779061, DOI https://doi.org/10.1090/S0002-9947-1985-0779061-2
- Brian Harbourne, Anticanonical rational surfaces, Trans. Amer. Math. Soc. 349 (1997), no. 3, 1191–1208. MR 1373636, DOI https://doi.org/10.1090/S0002-9947-97-01722-4
- André Hirschowitz, La méthode d’Horace pour l’interpolation à plusieurs variables, Manuscripta Math. 50 (1985), 337–388 (French, with English summary). MR 784148, DOI https://doi.org/10.1007/BF01168836
- André Hirschowitz, Une conjecture pour la cohomologie des diviseurs sur les surfaces rationnelles génériques, J. Reine Angew. Math. 397 (1989), 208–213 (French). MR 993223, DOI https://doi.org/10.1515/crll.1989.397.208
- D. Mumford, Some footnotes to the work of C. P. Ramanujam, C. P. Ramanujam—a tribute, Tata Inst. Fund. Res. Studies in Math., vol. 8, Springer, Berlin-New York, 1978, pp. 247–262. MR 541025
- Masayoshi Nagata, On the $14$-th problem of Hilbert, Amer. J. Math. 81 (1959), 766–772. MR 105409, DOI https://doi.org/10.2307/2372927
- Masayoshi Nagata, On rational surfaces. II, Mem. Coll. Sci. Univ. Kyoto Ser. A. Math. 33 (1960/61), 271–293. MR 126444, DOI https://doi.org/10.1215/kjm/1250775912
- Ulf Persson, On degenerations of algebraic surfaces, Mem. Amer. Math. Soc. 11 (1977), no. 189, xv+144. MR 466149, DOI https://doi.org/10.1090/memo/0189
- J. Roè: Limit linear systems and applications, preprint (2006); http://xxx.lanl.gov/ abs/math/0602213.
- Beniamino Segre, Alcune questioni su insiemi finiti di punti in geometria algebrica., Atti Convegno Internaz. Geometria Algebrica (Torino, 1961) Rattero, Turin, 1962, pp. 15–33 (Italian). MR 0146714
References
- G. Castelnuovo: Ricerche generali sopra i sistemi lineari di curve piane, Mem. Accad. Sci. Torino, II 42 (1891).
- C. Ciliberto, F. Cioffi, R. Miranda and F. Orecchia: Hermite interpolation and linear systems of plane curves with base fat points, Proceedings of ASCM03, World Sci. Publ. Co., Singapore, (2003), 87–102. MR 2061827 (2005c:41002)
- C. Ciliberto and R. Miranda: Degenerations of Planar Linear Systems, J. Reine Angew. Math. 501 (1998), 191–220. MR 1637857 (2000m:14005)
- C. Ciliberto and R. Miranda: Linear Systems of Plane Curves with Base Points of Equal Multiplicity, Trans. Amer. Math. Soc. 352 (2000), 4037–4050. MR 1637062 (2000m:14006)
- C. Ciliberto and R. Miranda: Matching conditions for degenerating plane curves and applications. In: Projective Varieties with Unexpected Properties, Proceedings of the Siena Conference, C. Ciliberto, A. V. Geramita, B. Harbourne, R. M. Miró-Roig, K. Ranestad ed., W. de Gruyter, 2005, 177–198. MR 2202252 (2006m:14044)
- C. Ciliberto and R. Miranda: The Segre and Harbourne–Hirschowitz Conjectures, in: Applications of algebraic geometry to coding theory, physics and computation (Eilat 2001), NATO Sci. Ser. II Math. Phys. Chem., 36, Kluwer Acad. Publ., Dordrecht, (2001), 37–51. MR 1866894 (2002k:14011)
- C. Ciliberto and R. Miranda: Nagata’s Conjecture for a Square Number of Points, Ricerche di Matematica, 55 (2006), 71–78. MR 2248163 (2007d:14098)
- T. de Fernex: Negative curves on very general blow-ups of $\mathbb {P}^ 2$, In: Projective Varieties with Unexpected Properties, Proceedings of the Siena Conference, C. Ciliberto, A. V. Geramita, B. Harbourne, R. M. Miró-Roig, K. Ranestad ed., W. de Gruyter, 2005, 199–207. MR 2202253 (2006k:14018)
- M. Dumnicki: Reduction method for linear systems of plane curves with base fat points. arXiv:math.AG/0606716, 2006.
- M. Dumnicki and W. Jarnicki: New effective bounds on the dimension of a linear system in $\mathbb {P}^2$, Journal of Symbolic Computation archive, Volume 42, Issue 6 (2007), 621–635. MR 2325918 (2008c:14009)
- L. Evain: Computing limit linear series with infinitesimal methods, Ann. Inst. Fourier (Grenoble) 57 (207), 1947-1974. MR 2377892 (2009e:14044)
- R. Friedman: Global smoothings of varieties with normal crossings, Ann. Math., 118 (1983), 75–114. MR 707162 (85g:32029)
- A. Gimigliano: On linear systems of plane curves, Thesis, Queen’s University, Kingston (1987).
- B. Harbourne: The geometry of rational surfaces and Hilbert functions of points in the plane, Proccedings of the 1984 Vancouver Conference in Algebraic Geometry, CMS Conf. Proc., 6, Amer. Math. Soc., Providence, RI (1986), 95–111. MR 846019 (87k:14041)
- B. Harbourne: Complete linear systems on rational surfaces, Trans. Amer, Math. Soc. 289 (1985), 213–226. MR 779061 (86h:14030)
- B. Harbourne: Anticanonical rational surfaces, Trans. AMS, 349 (1997), 1191–1208. MR 1373636 (97f:14007)
- A. Hirschowitz: La méthode d’Horace pour l’interpolation à plusieurs variables, Manuscripta Math., 50 (1985), 337–388. MR 784148 (86j:14013)
- A. Hirschowitz: Une conjecture pour la cohomologie des diviseurs sur les surfaces rationelles génériques, J. Reine Angew. Math. 397 (1989), 208–213. MR 993223 (90g:14021)
- D. Mumford: Some footnotes to the work of C. P. Ramanujam, in “C. P. Ramanujam – A tribute”, Tata Institute of Fundamental Research Studies in Mathematics (Springer Verlag) 8, 1978, 247–262. MR 541025 (80m:14026)
- M. Nagata: On the 14–th problem of Hilbert, Amer. J of Math. 81 (1959), 766–772. MR 0105409 (21:4151)
- M. Nagata: On rational surfaces II, Mem. Coll. Sci. Univ. Kyoto, Ser. A, Math. 33 (1960), 271, 293. MR 0126444 (23:A3740)
- U. Persson: On degenerations of algebraic surfaces, Memoirs of the AMS, Vol. 11, No. 189 (July, 1977). MR 0466149 (57:6030)
- J. Roè: Limit linear systems and applications, preprint (2006); http://xxx.lanl.gov/ abs/math/0602213.
- B. Segre: Alcune questioni su insiemi finiti di punti in geometria algebrica, Atti Convegno Intern. di Geom. Alg. di Torino (1961), 15–33. MR 0146714 (26:4234)
Additional Information
Ciro Ciliberto
Affiliation:
Dipartimento di Matematica, Universitá di Roma “Tor Vergata”, Via della Ricerca Scientifica, 00133 Roma, Italy
MR Author ID:
49480
Email:
cilibert@mat.uniroma2.it
Rick Miranda
Affiliation:
Department of Mathematics, Colorado State University, Fort Collins, Colorado 80523
Email:
rick.miranda@colostate.edu
Received by editor(s):
December 16, 2008
Received by editor(s) in revised form:
July 19, 2009
Published electronically:
February 8, 2011