Laudal’s lemma in positive characteristic
Author:
Paola Bonacini
Journal:
J. Algebraic Geom. 18 (2009), 459-475
DOI:
https://doi.org/10.1090/S1056-3911-08-00501-8
Published electronically:
July 3, 2008
MathSciNet review:
2496454
Full-text PDF
Abstract |
References |
Additional Information
Abstract: Laudal’s lemma states that if $C$ is a curve of degree $d>s^2+1$ in $\mathbb {P}^3$ over an algebraically closed field of characteristic $0$ such that its plane section is contained in an irreducible curve of degree $s$, then $C$ lies on a surface of degree $s$. We show that the same result does not hold in positive characteristic and we find different bounds $d>f(s)$ which ensure that $C$ is contained in a surface of degree $s$.
References
- W. Barth, Some properties of stable rank-$2$ vector bundles on ${\bf P}_{n}$, Math. Ann. 226 (1977), no. 2, 125–150. MR 429896, DOI https://doi.org/10.1007/BF01360864
- P. Bonacini, On the plane section of an integral curve in positive characteristic, to appear in Proc. A.M.S.
- Lawrence Ein, Stable vector bundles on projective spaces in ${\rm char}$ $p>0$, Math. Ann. 254 (1980), no. 1, 53–72. MR 597836, DOI https://doi.org/10.1007/BF01457885
- David Eisenbud, Commutative algebra, Graduate Texts in Mathematics, vol. 150, Springer-Verlag, New York, 1995. With a view toward algebraic geometry. MR 1322960
- A. V. Geramita, M. Roggero, and P. Valabrega, Subcanonical curves with the same postulation as $Q$ skew complete intersections in projective $3$-space, Istit. Lombardo Accad. Sci. Lett. Rend. A 123 (1989), 111–121 (1990) (English, with Italian summary). MR 1111356
- Laurent Gruson and Christian Peskine, Section plane d’une courbe gauche: postulation, Enumerative geometry and classical algebraic geometry (Nice, 1981), Progr. Math., vol. 24, Birkhäuser, Boston, Mass., 1982, pp. 33–35 (French). MR 685762
- Joe Harris, The genus of space curves, Math. Ann. 249 (1980), no. 3, 191–204. MR 579101, DOI https://doi.org/10.1007/BF01363895
- Robin Hartshorne, Algebraic geometry, Springer-Verlag, New York-Heidelberg, 1977. Graduate Texts in Mathematics, No. 52. MR 0463157
- Robin Hartshorne, Stable vector bundles of rank $2$ on ${\bf P}^{3}$, Math. Ann. 238 (1978), no. 3, 229–280. MR 514430, DOI https://doi.org/10.1007/BF01420250
- Robin Hartshorne, Stable reflexive sheaves, Math. Ann. 254 (1980), no. 2, 121–176. MR 597077, DOI https://doi.org/10.1007/BF01467074
- O. A. Laudal, A generalized trisecant lemma, Algebraic geometry (Proc. Sympos., Univ. Tromsø, Tromsø, 1977) Lecture Notes in Math., vol. 687, Springer, Berlin, 1978, pp. 112–149. MR 527232
- James S. Milne, Étale cohomology, Princeton Mathematical Series, No. 33, Princeton University Press, Princeton, N.J., 1980. MR 559531
- Rosario Strano, On generalized Laudal’s lemma, Complex projective geometry (Trieste, 1989/Bergen, 1989) London Math. Soc. Lecture Note Ser., vol. 179, Cambridge Univ. Press, Cambridge, 1992, pp. 284–293. MR 1201390, DOI https://doi.org/10.1017/CBO9780511662652.021
- G. Pete Wever, The moduli of a class of rank $2$ vector bundles on $P^{3}$, Nagoya Math. J. 84 (1981), 9–30. MR 641146
References
- W. P. Barth, Some properties of stale rank-$2$ vector bundles on $\mathbb P^n$, Math. Ann. 226 no. 2 (1977), 125–150. MR 0429896 (55:2905)
- P. Bonacini, On the plane section of an integral curve in positive characteristic, to appear in Proc. A.M.S.
- L. Ein, Stable vector bundles on projective spaces in $\emph {\text {char}} p>0$, Math. Ann. 254 (1980), 53–72. MR 597836 (82d:14010)
- D. Eisenbud, Commutative Algebra with a view toward Algebraic Geometry, Graduate Texts in Mathematics, no. 150, Springer-Verlag, New York, 1995. MR 1322960 (97a:13001)
- A. Geramita, M. Roggero, P. Valabrega, Subcanonical curves with the same postulation as $Q$ skew complete intersections in projective 3-space, Istit. Lombardo Accad. Sci. Lett. Rend. A 123 (1989), 111–121 (1990). MR 1111356 (92e:14026)
- L. Gruson, C. Peskine, Section plane d’une courbe gauche: postulation, Enumerative Geometry and Classical Algebraic Geometry (\uppercase{N}ice, 1981), Progress in Math., no. 24, Birkhäuser, Boston, Mass., 1982, pp. 33–35. MR 685762 (84c:14045)
- J. Harris, The Genus of space curves, Math. Ann. 249 (1980), 191–204. MR 579101 (81i:14022)
- R. Hartshorne, Algebraic Geometry, Graduate Texts in Mathematics, no. 52, Springer Verlag, New York, 1977. MR 0463157 (57:3116)
- ---, Stable vector bundles of rank 2 on $\mathbb P^3$, Math. Ann. 238 3 (1978), 229–280. MR 514430 (80c:14011)
- ---, Stable reflexive sheaves, Math. Ann. 254 2 (1980), 121–176. MR 597077 (82b:14011)
- O. A. Laudal, A generalized trisecant lemma, Algebraic Geometry (Proc. Sympos. Univ. Tromsø, Tromsø, 1977), Lecture Notes in Math., no. 687, Springer-Verlag, Berlin, 1978, pp. 112–149. MR 527232 (81f:14019)
- J. S. Milne, \uppercase{é}tale cohomology, Princeton Mathematical Series, no. 33, Princeton University Press, Princeton, N.J., 1980. MR 559531 (81j:14002)
- R. Strano, On generalized Laudal’s lemma, Complex projective geometry (Trieste 1989/Bergen 1989), London Math. Soc. Lecture Note Ser., no. 179, Cambridge Univ. Press, Cambridge, 1922, pp. 284–293. MR 1201390 (94a:14032)
- G. P. Wever, The moduli of a class of rank $2$ vector bundles on $\mathbb {P}^n$, Nagoya Math. J., 84 (1981), 9–30. MR 641146 (83c:14014)
Additional Information
Paola Bonacini
Affiliation:
University of Catania, Viale A. Doria 6, 95124, Catania, Italy
Email:
bonacini@dmi.unict.it
Received by editor(s):
January 26, 2007
Received by editor(s) in revised form:
September 30, 2007
Published electronically:
July 3, 2008
