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International Mathematics Research Papers (2008) Vol. 2008 : article ID rpn001, 65 pages, doi:10.1093/imrp/rpn001 published on April 1, 2008
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© The Author 2008. Published by Oxford University Press. All rights reserved. For Permissions, please e-mail: journals.permissions@oxfordjournals.org

Limits of Special Weierstrass Points

C. Cumino1, E. Esteves2 and L. Gatto1

1 Dipartimento di Matematica, Politecnico di Torino, C.so Duca degli Abruzzi 24, 10129 Torino, Italy
2 Instituto Nacional de Matemática Pura e Aplicada, Estrada Dona Castorina 110, 22460-320 Rio de Janeiro, Brazil

Correspondence: Correspondence to be sent to: IMPA, Estrada Dona Castorina 110, 22460-320 Rio de Janeiro, Brazil. e-mail: esteves{at}impa.br

Let C be the union of two general connected, smooth, nonrational curves X and Y intersecting transversally at a point P. Assume that P is a general point of X or of Y. Our main result describes all limits on C of special Weierstrass points along smooth curves degenerating to C. As an application, we recover in a unified and conceptually simpler way the computations made by Diaz and Cukierman of divisor classes of curves with special Weierstrass points in the moduli space of stable curves. In our approach there are no multiplicity issues, an usual nuisance of the method of test curves.



References

  1. Arbarello E., Cornalba M. The Picard groups of the moduli spaces of curves. Topology (1987) 26(2):153–71.[CrossRef][ISI]
  2. CoCoATeam. CoCoA: A system for doing Computations in Commutative Algebra. http://cocoa.dima.unige.it.
  3. Cukierman F. Families of Weierstrass points. Duke Mathematical Journal (1989) 58(2):317–46.[CrossRef][ISI]
  4. Cumino C., Esteves E., Gatto L. Special ramification loci on the double product of a general curve. Quarterly Journal of Mathematics (2007) preprint arXiv:math.AG/0701662.
  5. Diaz S. Tangent spaces in moduli via deformation with applications to Weierstrass points. Duke Mathematical Journal (1984) 51(4):905–22.[CrossRef][ISI]
  6. Diaz S. Exceptional Weierstrass Points and the Divisor on Moduli Space that they Define (1985) Memoires of the American Mathematical Society 56, no. 327. Providence, RI: American Mathematical Society.
  7. Eisenbud D., Harris J. Limit linear series: Basic theory. Inventiones Mathematicae (1986) 85(2):337–71.[CrossRef][ISI]
  8. Eisenbud D., Harris J. Existence, decomposition, and limits of certain Weierstrass points. Inventiones Mathematicae (1987) 87(3):495–515.[CrossRef][ISI]
  9. Esteves E. Wronski algebra systems on families of singular curves. Annales Scientifiques de l'École Normale Supérieure (4) (1996) 29(1):107–34.
  10. Esteves E. "Linear systems and ramification points on reducible nodal curves." Special issue. Matemática Contemporanea (1998) 14:21–35.
  11. Esteves E., Gatto L. A geometric interpretation and a new proof of a relation by Cornalba and Harris. Communications in Algebra (2003) 31(8):3753–70.[CrossRef][ISI]
  12. Esteves E., Medeiros N. Limit canonical systems on curves with two components. Inventiones Mathematicae (2002) 149(2):267–338.[CrossRef][ISI]
  13. Fulton W. Intersection Theory (1984) vol. 3. Ergebnisse der Mathematik und ihrer Grenzgebiete. Berlin: Springer.
  14. Gatto L. Weight sequences versus gap sequences at singular points of Gorenstein curves. Geometriae Dedicata (1995) 54(3):267–300.[CrossRef][ISI]
  15. Gatto L. On the closure in Formula of smooth curves having a special Weierstrass point. Mathematica Scandinavica (2001) 88(1):41–71.[ISI]
  16. Gieseker D. Lectures on Moduli of Curves. Tata Institute of Fundamental Research, Lecture Notes. (1982) Heidelberg, Germany: Springer.
  17. Harris J., Morrison I. Moduli of Curves. (1998) New York: Springer. Graduate Texts in Mathematics 187.
  18. Harris J., Mumford D. On the Kodaira dimension of the moduli space of curves. Inventiones Mathematicae (1982) 67:23–86.[CrossRef][ISI]
  19. Laksov D., Thorup A. The algebra of jets. Michigan Mathematical Journal (2000) 48:393–416.[CrossRef]
  20. Laksov D., Thorup A. Wronski systems for families of local complete intersection curves. Communications in Algebra (2003) 31(8):4007–35.[CrossRef][ISI]
  21. Looijenga E. Smooth Deligne–Mumford compactifications by means of Prym level structures. Journal of Algebraic Geometry (1994) 3:283–93.
  22. Mumford D. Stability of projective varieties. L'Enseignement Mathématique 2 (1977) 23:39–110.
  23. Mumford D. Towards an enumerative geometry of the moduli space of curves. In: Arithmetic and Geometry—Artin M., Tate J., eds. (1983) 2:271–328. Progress in Mathematics 36. Boston: Birkhäuser.
  24. Mumford D., Fogarty J. Geometric Invariant Theory (1982) 2nd ed. Berlin: Springer.
  25. Osserman B. A limit linear series moduli scheme. Annales de l'Institute Fourier (Grenoble) (2006) 56(4):1165–205.
  26. Pikaart M., Jong A. J. de. Moduli of curves with non-abelian level structure. The Moduli Space of Curves, 483–509. Progress in Mathematics 129. Boston: Birkhäuser, 1995.

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This Article
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Right arrow Articles by Gatto, L.
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