On a class of special linear systems of ℙ³

Laface, Antonio; Ugaglia, Luca

doi:10.1090/S0002-9947-06-03891-8

On a class of special linear systems of $\mathbb {P}^3$
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Trans. Amer. Math. Soc. 358 (2006), 5485-5500 Request permission

Abstract:

In this paper we deal with linear systems of $\mathbb {P}^3$ through fat points. We consider the behavior of these systems under a cubic Cremona transformation that allows us to produce a class of special systems which we conjecture to be the only ones.

References

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
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 10.1090/S0002-9947-00-02416-8
Cindy De Volder and Antonio Laface, On linear systems of $\mathbb {P}^3$ through multiple points, preprint.
William Fulton, Intersection theory, 2nd ed., 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. 2, Springer-Verlag, Berlin, 1998. MR 1644323, DOI 10.1007/978-1-4612-1700-8
G.-M. Greuel, G. Pfister, and H. Schönemann, Singular 2.0, A Computer Algebra System for Polynomial Computations, Centre for Computer Algebra, University of Kaiserslautern, 2001, http://www.singular.uni-kl.de.
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
Antonio Laface and Luca Ugaglia, A counterexample to a conjecture on linear systems on $\Bbb P^3$, Adv. Geom. 4 (2004), no. 3, 365–371. MR 2071811, DOI 10.1515/advg.2004.021
Evain Laurent, La fonction de Hilbert de la réunion de $4^h$ gros points génériques de $\textbf {P}^2$ de même multiplicité, J. Algebraic Geom. 8 (1999), no. 4, 787–796 (French, with French summary). MR 1703614
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