Blowup and specialization methods for the study of linear systems

Joaquim Roé


The computation of the dimension of linear systems of curves with imposed base multiple points on surfaces is a difficult problem, with open conjectures that are being approached only with partial success. Among others, blowup-based techniques and degenerations show some promise of leading to satisfactory answers. We present an overview of such blowup based techniques at an introductory level, with emphasis on clusters of infinitely near points and Ciliberto-Miranda's blowup and twist.


blowup, twist

Mathematics Subject Classification

14C20, 14D06, (32C10, 14C05, 14E05)


Alexander, J. and A. Hirschowitz. "An asymptotic vanishing theorem for generic unions of multiple points." Invent. Math. 140 (2000): 303-325.

---. "Un lemme d’Horace différentiel: application aux singularités hyperquartiques de P5." J. Algebraic Geom. 1 (1992): 411-426.

Barth, W.P. et al. "Compact complex surfaces." Second ed. Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge. A Series of Modern Surveys in Mathematics 4, Berlin: Springer-Verlag, 2004.

Bauer, T. et al. "A primer on Seshadri constants." Interactions of classical and numerical algebraic geometry. Contemp. Math. 496. Providence, RI: Amer. Math. Soc., 2009. 33-70.

Beauville, A. "Complex algebraic surfaces." London Mathematical Society Student Texts 34. Cambridge: Cambridge University Press, 1996.

Casas-Alvero, E. "Singularities of plane curves." London Mathematical Society Lecture Notes Series 276. Cambridge: Cambridge University Press, 2000.

Ciliberto, C. and R. Miranda. "Degenerations of planar linear systems." J. Reine Angew. Math. 501 (1998): 191-220.

---. "Homogeneous interpolation on ten points." J. Algebraic Geom. 20 (2011): 685-726.

---. "Linear systems of plane curves with base points of equal multiplicity." Trans. Amer. Math. Soc. 352 (2000): 4037-4050.

---. "Matching conditions for degenerating plane curves and applications." Projective varieties with unexpected properties. Berlin: Walter de Gruyter GmbH & Co. KG, 2005. 177-197.

---. "The Segre and Harbourne-Hirschowitz conjectures." NATO Sci. Ser. II Math. Phys. Chem. 36. Publ., Dordrecht: Kluwer Acad., 2001. 37-51.

Demailly, J.P. "Singular Hermitian metrics on positive line bundles." Lecture Notes in Math. 1507. Berlin: Springer, 1992. 87-104.

Douady, A. "Le problème des modules pour les sous-espaces analytiques compacts d’un espace analytique donné." Ann. Inst. Fourier (Grenoble), 16 (1966): 1-95.

Dumnicki, M. "Cutting diagram method for systems of plane curves with base points." Ann. Polon. Math. 90 (2007): 131-143.

Enriques, F. and O. Chisini. "Lezioni sulla teoria geometrica delle equazioni e delle funzioni algebriche." Mathematics Collection 5. Bologna: Nicola Zanichelli Editore S.p.A., 1985.

Évain, L. "Collisions de trois gros points sur une surface algébrique." Ph.D. thesis. Université de Nice, 1997.

---. "Computing limit linear series with infinitesimal methods." Ann. Inst. Fourier (Grenoble) 57 (2007): 1947-1974.

Fulton, W. "Algebraic curves." Reprint of 1969 original. Advanced Book Classics. Redwood City, CA: Addison-Wesley Publishing Company, 1989.

---. "Intersection theory." Second edition. Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge. A Series of Modern Surveys in Mathematics 2. Berlin: Springer-Verlag, 1998.

Geramita, A.V., B. Harbourne and J. Migliore. "Classifying Hilbert functions of fat point subschemes in P2." Collect. Math. 60 (2009): 159-192.

Gimigliano, A. "On linear systems of plane curves." Ph.D. thesis. Queen’s University (Canda), 1987.

Harbourne, B. "Complete linear systems on rational surfaces." Trans. Amar. Math. Soc. 289 (1985): 213-226.

---. "The geometry of rational surfaces and the Hilbert functions of points in the plane." Can. Math. Soc. Conf. Proc. 6 (1986): 95-111.

Harbourne, B. and J. Roé. "Discrete behavior of seshadri constants on surfaces." J. Pure Appl. Algebra, 212 (2008): 616-627.

---. "Linear systems with multiple base points in P2." Adv. Geom. 4 (2004): 41-59.

Hartshorne, R. "Algebraic geometry." Graduate Texts in Mathematics 52. New York-Heidelberg: Springer-Verlag, 1977.

Hirschowitz, A. "La méthode d’Horace pour l’interpolation a plusieurs variables." Manuscripta Math. 50 (1985): 337-388.

---. "Une conjecture pour la cohomologie des diviseurs sur les surfaces rationelles génériques." J. Reine Angew. Math. 397 (1989): 208-213.

Kleiman, S.L. "Multiple-point formulas I. Iteration." Acta Math. 147 (1981): 13-49.

---. "Toward a numerical theory of ampleness" Ann. of Math. (2) 84 (1966): 293-344.

Lazarsfeld, R. "Positivity in algebraic geometry. I, Classical setting: line bundles and linear series." Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge. A Series of Modern Surveys in Mathematics 48. Berlin: Springer- Verlag, 2004.

Lejeune-Jalabert, M. and B. Teissier. "Normal cones and sheaves of relative jets." Compositio Math. 28 (1974): 305-331.

McDuff, D. and L. Polterovich. "Symplectic packings and algebraic geometry." Invent. Math. 115 (1994): 405-434.

Monserrat, F. "Curves having one place at infinity and linear systems on rational surfaces, J. Pure Appl. Algebra, 211 (2007), 685–701.

Mumford, D. "Lectures on curves on an algebraic surface." Annals of Mathematics Studies 59. Princeton, N.J.: Princeton University Press, 1966.

Nagata, M. "On the fourteenth problem of Hilbert." Amer. J. Math. 81 (1959): 766-772.

Nitsure, N. "Construction of Hilbert and Quot schemes, Fundamental algebraic geometry." Math. Surveys Monogr. 123. Providence, RI: Amer. Math. Soc., 2005. 105-137.

Oguiso, K. "Seshadri constants in a family of surfaces." Math. Ann. 323 (2002): 625-631.

Roé, J. "A relation between one-point and multi-point Seshadri constants." J. Algebra, 274 (2004): 643-651.

---. "Limit linear systems and applications." preprint (2006/2009). arXiv:math/0602213.

---. "Maximal rank for planar singularities of multiplicity 2." J. Algebra 302 (2006): 37-54.

---. "On the conditions imposed by tacnodes and cusps." Trans. Amer. Math. Soc. 353 (2001): 4925-4948.

---. "On the existence of plane curves with imposed multiple points." J. Pure Appl. Alg. 156 (2001): 115-126.

---. "Varieties of clusters and Enriques diagrams." Math. Proc. Cambridge Philos. Soc. 137 (2004): 69-94.

Roé, J. and J. Ross. "An inequality between multipoint Seshadri constants." Geom. Dedicata 140 (2009): 175-181.

Segre, B. "Alcune questioni su insiemi finiti di punti in geometria algebrica." Univ. e Politec. Torino Rend. Sem. Mat. 20 (1960/1961): 67-85.

Szemberg, T. "Global and local positivity of line bundles." Habilitation. Essen, 2001.

Walker, R.J. "Algebraic curves" Reprint of the 1950 edition. New York-Heidelberg: Springer-Verlag, 1978.

Zariski, O. "Algebraic surfaces." Second supplemented edition. Ergebnisse der Mathematik und ihrer Grenzgebiete 61. New York-Heidelberg: Springer-Verlag, 1971.

Zariski, O. and P. Samuel. "Commutative algebra. Vol. II." Graduate Texts in Mathematics 29. New York-Heidelberg: Springer-Verlag, 1975.

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