General blow-ups of the projective plane

Szemberg, Tomasz; Tutaj-Gasińska, Halszka

doi:10.1090/S0002-9939-02-06488-2

General blow-ups of the projective plane
HTML articles powered by AMS MathViewer

by Tomasz Szemberg and Halszka Tutaj-Gasińska PDF

Proc. Amer. Math. Soc. 130 (2002), 2515-2524 Request permission

Abstract:

We study linear series on a projective plane blown up in a bunch of general points. Such series arise from plane curves of fixed degree with assigned fat base points. We give conditions (expressed as inequalities involving the number of points and the degree of the plane curves) on these series to be base point free, i.e. to define a morphism to a projective space. We also provide conditions for the morphism to be a higher order embedding. In the discussion of the optimality of obtained results we relate them to the Nagata Conjecture expressed in the language of Seshadri constants and we give a lower bound on these invariants.

References

Edoardo Ballico and Marc Coppens, Very ample line bundles on blown-up projective varieties, Bull. Belg. Math. Soc. Simon Stevin 4 (1997), no. 4, 437–447. MR 1476035
Thomas Bauer, Seshadri constants on algebraic surfaces, Math. Ann. 313 (1999), no. 3, 547–583. MR 1678549, DOI 10.1007/s002080050272
M. Beltrametti and A. J. Sommese, Zero cycles and $k$th order embeddings of smooth projective surfaces, Problems in the theory of surfaces and their classification (Cortona, 1988) Sympos. Math., XXXII, Academic Press, London, 1991, pp. 33–48. With an appendix by Lothar Göttsche. MR 1273371
Ekin M. Bese, On the spannedness and very ampleness of certain line bundles on the blow-ups of $\textbf {P}^{2}_{\textbf {C}}$ and $F_{r}$, Math. Ann. 262 (1983), no. 2, 225–238. MR 690197, DOI 10.1007/BF01455313
Aldo Biancofiore, On the hyperplane sections of blow-ups of complex projective plane, Canad. J. Math. 41 (1989), no. 6, 1005–1020. MR 1018450, DOI 10.4153/CJM-1989-045-5
Paul Biran, Constructing new ample divisors out of old ones, Duke Math. J. 98 (1999), no. 1, 113–135. MR 1687571, DOI 10.1215/S0012-7094-99-09803-4
Fabrizio Catanese and Lothar Gœttsche, $d$-very-ample line bundles and embeddings of Hilbert schemes of $0$-cycles, Manuscripta Math. 68 (1990), no. 3, 337–341. MR 1065935, DOI 10.1007/BF02568768
Marc Coppens, Embeddings of general blowing-ups at points, J. Reine Angew. Math. 469 (1995), 179–198. MR 1363829, DOI 10.1515/crll.1995.469.179
Jean d’Almeida and André Hirschowitz, Quelques plongements projectifs non spéciaux de surfaces rationnelles, Math. Z. 211 (1992), no. 3, 479–483 (French). MR 1190223, DOI 10.1007/BF02571440
Jean-Pierre Demailly, Singular Hermitian metrics on positive line bundles, Complex algebraic varieties (Bayreuth, 1990) Lecture Notes in Math., vol. 1507, Springer, Berlin, 1992, pp. 87–104. MR 1178721, DOI 10.1007/BFb0094512
Sandra Di Rocco, $k$-very ample line bundles on del Pezzo surfaces, Math. Nachr. 179 (1996), 47–56. MR 1389449, DOI 10.1002/mana.19961790104
Lawrence Ein and Robert Lazarsfeld, Seshadri constants on smooth surfaces, Astérisque 218 (1993), 177–186. Journées de Géométrie Algébrique d’Orsay (Orsay, 1992). MR 1265313
Brian Harbourne, On Nagata’s conjecture, J. Algebra 236 (2001), no. 2, 692–702. MR 1813496, DOI 10.1006/jabr.2000.8515
—, Seshadri constants and very ample divisors on algebraic surfaces. Preprint alg-geom 0103029.
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 10.1515/crll.1989.397.208
Oliver Küchle, Ample line bundles on blown up surfaces, Math. Ann. 304 (1996), no. 1, 151–155. MR 1367887, DOI 10.1007/BF01446289
Thierry Mignon, Systèmes de courbes planes à singularités imposées: le cas des multiplicités inférieures ou égales à quatre, J. Pure Appl. Algebra 151 (2000), no. 2, 173–195 (French, with English summary). MR 1775572, DOI 10.1016/S0022-4049(99)00054-7
Masayoshi Nagata, On the $14$-th problem of Hilbert, Amer. J. Math. 81 (1959), 766–772. MR 105409, DOI 10.2307/2372927
Oguiso, K., Seshadri constant for a family of surfaces. Preprint alg-geom 0002211.
H. Tutaj-Gasińska, Embeddings of blown-up plane, Geom. Dedicata 76 (1999), no. 2, 165–175. MR 1703211, DOI 10.1023/A:1005106123696
Geng Xu, Curves in $\textbf {P}^2$ and symplectic packings, Math. Ann. 299 (1994), no. 4, 609–613. MR 1286887, DOI 10.1007/BF01459801
Geng Xu, Divisors on the blow-up of the projective plane, Manuscripta Math. 86 (1995), no. 2, 195–197. MR 1317744, DOI 10.1007/BF02567988
Geng Xu, Ample line bundles on smooth surfaces, J. Reine Angew. Math. 469 (1995), 199–209. MR 1363830, DOI 10.1515/crll.1995.469.199