Special systems through double points on an algebraic surface

Laface, Antonio

doi:10.1090/S0002-9939-2011-10845-1

Special systems through double points on an algebraic surface
HTML articles powered by AMS MathViewer

by Antonio Laface

Proc. Amer. Math. Soc. 139 (2011), 1971-1981

DOI: https://doi.org/10.1090/S0002-9939-2011-10845-1

Published electronically: January 21, 2011

PDF | Request permission

Abstract:

Let $S$ be a smooth projective algebraic surface satisfying the following property: $H^i(S,B)=0$ for $i>0$, for any irreducible and reduced curve $B$ of $S$. The aim of this paper is to provide a characterization of special linear systems on $S$ which are singular along a set of double points in very general position. As an application, the dimension of such systems is evaluated in case $S$ is a simple Abelian surface, a $K3$ surface which does not contain elliptic curves or an anticanonical rational surface.

References

Arnaud Beauville, Complex algebraic surfaces, 2nd ed., London Mathematical Society Student Texts, vol. 34, Cambridge University Press, Cambridge, 1996. Translated from the 1978 French original by R. Barlow, with assistance from N. I. Shepherd-Barron and M. Reid. MR 1406314, DOI 10.1017/CBO9780511623936
V.A. Iskovskikh. Encyclopaedia of Mathematics: Bertini theorems. http://eom.springer.de/B/b015770.htm.
W. Barth, C. Peters, and A. Van de Ven, Compact complex surfaces, Ergebnisse der Mathematik und ihrer Grenzgebiete (3) [Results in Mathematics and Related Areas (3)], vol. 4, Springer-Verlag, Berlin, 1984. MR 749574, DOI 10.1007/978-3-642-96754-2
M. V. Catalisano, A. V. Geramita, and A. Gimigliano, Higher secant varieties of Segre-Veronese varieties, Projective varieties with unexpected properties, Walter de Gruyter, Berlin, 2005, pp. 81–107. MR 2202248
L. Chiantini and C. Ciliberto, Weakly defective varieties, Trans. Amer. Math. Soc. 354 (2002), no. 1, 151–178. MR 1859030, DOI 10.1090/S0002-9947-01-02810-0
Ciro Ciliberto, Geometric aspects of polynomial interpolation in more variables and of Waring’s problem, European Congress of Mathematics, Vol. I (Barcelona, 2000) Progr. Math., vol. 201, Birkhäuser, Basel, 2001, pp. 289–316. MR 1905326
M. Dale, Terracini’s lemma and the secant variety of a curve, Proc. London Math. Soc. (3) 49 (1984), no. 2, 329–339. MR 748993, DOI 10.1112/plms/s3-49.2.329
Cindy De Volder and Antonio Laface, Linear systems on generic $K3$ surfaces, Bull. Belg. Math. Soc. Simon Stevin 12 (2005), no. 4, 481–489. MR 2205992
Alessandro Gimigliano, ON LINEAR SYSTEMS OF PLANE CURVES, ProQuest LLC, Ann Arbor, MI, 1987. Thesis (Ph.D.)–Queen’s University (Canada). MR 2635606
Brian Harbourne, Very ample divisors on rational surfaces, Math. Ann. 272 (1985), no. 1, 139–153. MR 794097, DOI 10.1007/BF01455934
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
Robin Hartshorne, Algebraic geometry, Graduate Texts in Mathematics, No. 52, Springer-Verlag, New York-Heidelberg, 1977. MR 0463157
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
Robert Lazarsfeld, Positivity in algebraic geometry. I, Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge. A Series of Modern Surveys in Mathematics [Results in Mathematics and Related Areas. 3rd Series. A Series of Modern Surveys in Mathematics], vol. 48, Springer-Verlag, Berlin, 2004. Classical setting: line bundles and linear series. MR 2095471, DOI 10.1007/978-3-642-18808-4
Beniamino Segre, Alcune questioni su insiemi finiti di punti in geometria algebrica, Univ. e Politec. Torino Rend. Sem. Mat. 20 (1960/61), 67–85 (Italian). MR 169119
A. Terracini. Su due problemi, concernenti la determinazione di alcune classi di superficie, considerati da G. Scorza e F. Palatini. Atti Soc. Natur. e Matem. Modena, (6):3–16, 1921-22.