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

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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 $s$.

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*Some properties of stale rank-$2$ vector bundles on $\mathbb P^n$*, Math. Ann. **226** no. 2 (1977), 125–150. MR **0429896 (55:2905)**
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*On the plane section of an integral curve in positive characteristic*, to appear in Proc. A.M.S.
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- 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)**
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*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)**
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*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)**
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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