Journal of Algebraic Geometry

Author(s): Gian Pietro Pirola; Enrico Schlesinger
Journal: J. Algebraic Geom. 14 (2005), 623-642.
Posted: April 25, 2005
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Abstract: The uniform position principle states that, given an irreducible non- degenerate curve $C \subset \mathbb{P} ^r (\mathbb{C} )$ , a general

-plane $L \subset \mathbb{P} ^r$ is uniform; that is, projection from

induces a rational map $C \dashrightarrow \mathbb{P} ^{1}$ whose monodromy group is the full symmetric group. In this paper we first show the locus of non-uniform

-planes has codimension at least two in the Grassmannian. This result is sharp because, if there is a point $x \in \mathbb{P} ^r$ such that projection from

induces a map $C \dashrightarrow \mathbb{P} ^{r-1}$ that is not birational onto its image, then the Schubert cycle $\sigma(x)$ of

-planes through

is contained in the locus of non-uniform

-planes. For a smooth curve

in $\mathbb{P} ^3$ , we show that any irreducible surface of non-uniform lines is a cycle $\sigma(x)$ as above, unless

is a rational curve of degree three, four, or six.

1.: E. Arbarello, M. Cornalba, P. A. Griffiths, and J. Harris.
Geometry of algebraic curves. Vol. I.
Springer-Verlag, New York, 1985. MR 0770932 (86h:14019)
2.: M. Artebani and G. P. Pirola.
Algebraic functions with even monodromy.
Proc. Amer. Math. Soc. 133(2):331-341 (electronic), 2005. MR 2093052
3.: W. Barth and R. Moore.
On rational plane sextics with six tritangents.
In Algebraic geometry and commutative algebra, Vol. I, pages 45-58. Kinokuniya, Tokyo, 1988. MR 0977752 (90c:14018)
4.: F. Cukierman.
Monodromy of projections.
Mat. Contemp. 16:9-30, 1999.
15th School of Algebra (Portuguese) (Canela, 1998). MR 1756825 (2001b:14018)
5.: D. Eisenbud.
Linear sections of determinantal varieties.
Amer. J. Math. 110(3):541-575, 1988. MR 0944327 (89h:14041)
6.: W. Fulton and R. Lazarsfeld.
Connectivity and its applications in algebraic geometry.
In Algebraic geometry (Chicago, Ill., 1980), pages 26-92. Springer, Berlin, 1981. MR 0644817 (83i:14002)
7.: R. Guralnick and K. Magaard.
On the minimal degree of a primitive permutation group.
J. Algebra 207(1):127-145, 1998. MR 1643074 (99g:20014)
8.: J. Harris.
Galois groups of enumerative problems.
Duke Math. J. 46(4):685-724, 1979. MR 0552521 (80m:14038)
9.: J. Harris.
Curves in Projective Space.
Sem. de Mathématiques Superieures. Université de Montreal, 1982
(with the collaboration of D. Eisenbud). MR 0685427 (84g:14024)
10.: R. Hartshorne,
Algebraic Geometry GTM 52, Springer-Verlag, Berlin, Heidelberg and New York, 1977. MR 0463157 (57:3116)
11.: H. Kaji,
On the tangentially degenerate curves.
Journal of the London Math. Soc. (2) 33 (1986), no. 3, 430-440. MR 0850959 (87i:14027)
12.: K. Magaard and H. Völklein.
The monodromy group of a function on a general curve.
Israel J. Math. 141:355-368, 2004. MR 2063042 (2005e:14047)
13.: K. Miura.
Field theory for function fields of plane quintic curves.
Algebra Colloq. 9(3):303-312, 2002. MR 1917155 (2003e:14016)
14.: K. Miura and H. Yoshihara.
Field theory for function fields of plane quartic curves.
J. Algebra 226(1):283-294, 2000. MR 1749889 (2001f:14047a)
15.: M. V. Nori.
Zariski's conjecture and related problems.
Ann. Sci. École Norm. Sup. (4) 16(2):305-344, 1983. MR 0732347 (86d:14027)
16.: G. Pirola.
Algebraic curves and non-rigid minimal surfaces in the euclidean space,
Pacific J. Math. (183), no. 2, 333-357, 1998. MR 1625966 (99e:53009)
17.: R. Strano.
Hyperplane sections of reducible curves.
In Zero-dimensional schemes and applications (Naples, 2000), pages 55-62. Queen's Univ., Kingston, ON, 2002. MR 1898825 (2003c:14033)
18.: C. Voisin.
Théorie de Hodge et géométrie algébrique complexe, volume 10 of Cours Spécialisés.
Société Mathématique de France, Paris, 2002. MR 1988456 (2005c:32024a)
19.: H. Yoshihara.
Function field theory of plane curves by dual curves.
J. Algebra 239(1):340-355, 2001. MR 1827887 (2002f:14038)
20.: O. Zariski,
Sull'impossibilità di risolvere parametricamente per radicali un'equazione algebrica di genere a moduli generali.
Atti Accad. Naz. Lincei Rend., Cl. Sc. Fis. Mat. Natur., serie VI 3:660-666, 1926.
21.: O. Zariski,
A theorem on the Poincaré group of an algebraic hypersurface.
Ann. of Math. 38:131-142, 1937. MR 1503330

Gian Pietro Pirola
Affiliation: Dipartimento di Matematica ``F. Casorati'', Università di Pavia, via Ferrata 1, 27100 Pavia, Italia
Email: pirola@dimat.unipv.it

Enrico Schlesinger
Affiliation: Dipartimento di Matematica, Politecnico di Milano, Piazza Leonardo da Vinci 32, 20133 Milano, Italia
Email: enrsch@mate.polimi.it

PII: S 1056-3911(05)00408-X
Received by editor(s): January 21, 2004
Received by editor(s) in revised form: February 10, 2005
Posted: April 25, 2005
Additional Notes: The first author was partially supported by: 1) MIUR PRIN 2003: {Spazi di moduli e teoria di Lie}; 2) Gnsaga; 3) Far 2002 (PV): Varietà algebriche, calcolo algebrico, grafi orientati e topologici. The second author was partially supported by MIUR PRIN 2002 Geometria e classificazione delle varietà proiettive complesse.

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