Hostname: page-component-cd9895bd7-dk4vv Total loading time: 0 Render date: 2024-12-23T19:11:01.079Z Has data issue: false hasContentIssue false
  • English
  • Français

A Degree Formula for Secant Varieties of Curves

Published online by Cambridge University Press:  21 August 2013

Rüdiger Achilles ,
Mirella Manaresi  and
Peter Schenzel
Rüdiger Achilles
Affiliation:
Dipartimento de Matematica, Università di Bologna, Piazza di Porta San Donato 5, 40126 Bologna, Italy, ([email protected]; [email protected])
Mirella Manaresi
Affiliation:
Dipartimento de Matematica, Università di Bologna, Piazza di Porta San Donato 5, 40126 Bologna, Italy, ([email protected]; [email protected])
Peter Schenzel
Affiliation:
Institut für Informatik, Martin-Luther-Universität, Von-Seckendorff-Platz 1, 06120 Halle (Saale), Germany, ([email protected])
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

Using the Stückrad–Vogel self-intersection cycle of an irreducible and reduced curve in pro-jective space, we obtain a formula that relates the degree of the secant variety, the degree and the genus of the curve and the self-intersection numbers, the multiplicities and the number of branches of the curve at its singular points. From this formula we deduce an expression for the difference between the genera of the curve. This result shows that the self-intersection multiplicity of a curve in projective N-space at a singular point is a natural generalization of the intersection multiplicity of a plane curve with its generic polar curve. In this approach, the degree of the secant variety (up to a factor 2), the self-intersection numbers and the multiplicities of the singular points are leading coefficients of a bivariate Hilbert polynomial, which can be computed by computer algebra systems.

Keywords

curves in projective spaceself-intersection multiplicitiesarithmetic genusgeometric genussecant varietiesPrimary 14C1714E22Secondary 14H2014N15
Type
Research Article
Information
Proceedings of the Edinburgh Mathematical Society , Volume 57 , Issue 2 , June 2014 , pp. 305 - 322
Copyright
Copyright © Edinburgh Mathematical Society 2014 

References

1.Achilles, R. and Aliffi, D., Segre, a script for the Reduce package Cali, University of Bologna (2012) (available at www.dm.unibo.it/~achilles/segre/).Google Scholar
2.Achilles, R. and Manaresi, M., An algebraic characterization of distinguished varieties of intersection, Rev. Roumaine Math. Pures Appl. 38 (1993), 569–578.Google Scholar
3.Achilles, R. and Manaresi, M., Multiplicities of a bigraded ring and intersection theory, Math. Ann. 309 (1997), 573–591.CrossRefGoogle Scholar
4.Achilles, R. and Vogel, W., On multiplicities for improper intersections, J. Alg. 168 (1994), 123–142.CrossRefGoogle Scholar
5.Briançon, J., Galligo, A. and Granger, M., Déformations équisingidières des germes de courbes gauches réduites, Volume 1, p. 69 (Société Mathématiques de France, Paris, 1981).Google Scholar
6.Buchweitz, R.-O. and Greuel, G.-M., The Milnor number and deformations of complex curve singularities, Invent. Math. 58 (1980), 241–281.Google Scholar
7.Chądzyński, J., Krasiński, T. and Tworzewski, P., On the intersection multiplicity of analytic curves in ℂm, Bull. Polish Acad. Sci. Math. 45 (1997), 163–169.Google Scholar
8.Dale, M., Terracini's lemma and the secant variety of a curve, Proc. Lond. Math. Soc. 49 (1984), 329–339.Google Scholar
9.Fischer, G., Plane algebraic curves (American Mathematical Society, Providence, RI, 2001).Google Scholar
10.Flenner, H. and Manaresi, M., Intersection of projective varieties and generic projections, Manuscr. Math. 92 (1997), 273–286.Google Scholar
11.Flenner, H., O'Carroll, L. and Vogel, W., Joins and intersections, Springer Monographs in Mathematics (Springer, 1999).Google Scholar
12.Griffiths, Ph. and Harris, J., Principles of algebraic geometry, Pure and Applied Mathematics, Volume 52 (Wiley, 1978).Google Scholar
13.Harris, J., Algebraic geometry: a first course, Graduate Texts in Mathematics, Volume 133 (Springer, 1995).Google Scholar
14.Hartshorne, R., Algebraic geometry, Graduate Texts in Mathematics, Volume 52 (Springer, 1977).Google Scholar
15.Krasiński, T., Improper intersection of analytic curves, Univ. Iagel. Acta Math. 39 (2001), 153–166.Google Scholar
16.Lê, D. T., Computation of the Milnor number of an isolated singularity of a complete intersection, Funkcional. Analysis i Priložen 8 (1974), 45–49 (in Russian).Google Scholar
17.Mond, B. and Pellikaan, R., Fitting ideals and multiple points of analytic mappings, in Algebraic geometry and complex analysis (ed. de Arellano, R.), Lecture Notes in Mathematics, Volume 1414, pp. 107–161 (Springer, 1989).Google Scholar
18.Peters, C. A. M. and Simonis, J., A secant formula, Q. J. Math. 27 (1976), 181–189.Google Scholar
19.Ranestad, K., The degree of the secant variety and the join of monomial curves, Collectanea Math. 57 (2006), 27–41.Google Scholar
20.Rosenlicht, M., Equivalence relations on algebraic curves, Annals Math. 56 (1952), 169–191.Google Scholar
21.Semple, J. G. and Roth, L., Introduction to algebraic geometry, Oxford Science Publications (Clarendon, Oxford, 1985).Google Scholar
22.Serre, J.-P., Algebraic groups and class fields (Springer, 1988).CrossRefGoogle Scholar
23.Stückrad, J. and Vogel, W., An algebraic approach to the intersection theory, in The curves seminar at Queen's, Volume II, pp. 1–32, Queen's Papers in Pure and Applied Mathematics, Volume 61 (Queen's University, Kingston, ON, 1982).Google Scholar
24.Stückrad, J. and Vogel, W., An Euler-Poincare characteristic for improper intersections, Math. Ann. 274 (1986), 257–271.Google Scholar
25.Van Gastel, L. J., Excess intersections, PhD Thesis, Rijksuniversiteit Utrecht (1989).Google Scholar
26.Van Gastel, L. J., Excess intersections and a correspondence principle, Invent. Math. 103 (1991), 197–221.Google Scholar
27.Whitney, H., Local properties of analytic varieties, in Differential and Combinatorial Topology (A Symposium in Honor of Marston Morse), pp. 205–244 (Princeton University Press, Princeton, NJ, 1965).CrossRefGoogle Scholar