# Enumerative geometry of curves with exceptional secant planes

Abstract : We study curves with linear series that are exceptional with regard to their secant planes. Working in the framework of an extension of Brill-Noether theory to pairs of linear series, we prove that a general curve of genus g has no exceptional secant planes, in a very precise sense. We also address the problem of computing the number of linear series with exceptional secant planes in a one-parameter family in terms of tautological classes associated with the family. We obtain conjectural generating functions for the tautological coefficients of secant-plane formulas associated to series $g^{2d-1}_m$ that admit $d$-secant $(d-2)$-planes. We also describe a strategy for computing the classes of divisors associated to exceptional secant plane behavior in the Picard group of the moduli space of curves in a couple of naturally-arising infinite families of cases, and we give a formula for the number of linear series with exceptional secant planes on a general curve equipped with a one-dimensional family of linear series.
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Mathematics [math]. Harvard University, 2007. English
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https://tel.archives-ouvertes.fr/tel-00466017
Contributor : Ethan Cotterill <>
Submitted on : Monday, March 22, 2010 - 2:42:07 PM
Last modification on : Friday, March 27, 2015 - 10:01:05 AM

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• HAL Id : tel-00466017, version 1

### Citation

Ethan Cotterill. Enumerative geometry of curves with exceptional secant planes. Mathematics [math]. Harvard University, 2007. English. <tel-00466017>

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