Monodromy of projective curves

Authors:
Gian Pietro Pirola and Enrico Schlesinger

Journal:
J. Algebraic Geom. **14** (2005), 623-642

DOI:
https://doi.org/10.1090/S1056-3911-05-00408-X

Published electronically:
April 25, 2005

MathSciNet review:
2147355

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Abstract |
References |
Additional Information

Abstract: The uniform position principle states that, given an irreducible non- degenerate curve $C \subset \mathbb {P}^r (\mathbb {C})$, a general $(r\!-\!2)$-plane $L \subset \mathbb {P}^r$ is *uniform*; that is, projection from $L$ 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 $(r-2)$-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 $x$ induces a map $C \dashrightarrow \mathbb {P}^{r-1}$ that is not birational onto its image, then the Schubert cycle $\sigma (x)$ of $(r\!-\!2)$-planes through $x$ is contained in the locus of non-uniform $(r\!-\!2)$-planes. For a smooth curve $C$ in $\mathbb {P}^3$, we show that any irreducible surface of non-uniform lines is a cycle $\sigma (x)$ as above, unless $C$ is a rational curve of degree three, four, or six.

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Additional Information

**Gian Pietro Pirola**

Affiliation:
Dipartimento di Matematica “F. Casorati”, Università di Pavia, via Ferrata 1, 27100 Pavia, Italia

MR Author ID:
139965

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

Received by editor(s):
January 21, 2004

Received by editor(s) in revised form:
February 10, 2005

Published electronically:
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.