[Paper Review] Geometry of curves with exceptional secant planes

Ethan Cotterill|arXiv (Cornell University)|Jun 14, 2007

Algebraic Geometry and Number Theory30 references11 citations

TL;DR

This paper extends Brill-Noether theory to pairs of linear series to study curves with exceptional secant planes, proving that a general curve of genus g has no such planes. It derives tautological formulas for counting exceptional linear series in families and conjectures generating functions for d-secant (d−2)-planes, offering a framework for computing divisor classes in the moduli space of curves.

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 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. tautological coefficients of secant-plane formulas associated to series g 2d−1 m

Motivation & Objective

• To investigate curves with linear series that are exceptional with respect to their secant planes using an extended framework of Brill-Noether theory.
• To prove that a general curve of genus g admits no exceptional secant planes in a precise, geometric sense.
• To compute the number of linear series with exceptional secant planes in a one-parameter family using tautological classes.
• To develop a strategy for computing divisor classes associated with exceptional secant plane behavior in the Picard group of the moduli space of curves.
• To provide a formula for the number of exceptional linear series on a general curve equipped with a one-dimensional family of linear series.

Proposed method

• Extends Brill-Noether theory to pairs of linear series to analyze exceptional secant plane behavior.
• Uses tautological classes associated with one-parameter families to compute the number of exceptional linear series.
• Applies geometric and cohomological techniques to derive formulas for secant-plane configurations.
• Proposes conjectural generating functions for the number of d-secant (d−2)-planes via tautological coefficients.
• Develops a strategy for computing divisor classes in the Picard group of the moduli space of curves in infinite families of cases.
• Employs intersection theory and moduli-theoretic methods to relate linear series behavior to tautological classes.

Experimental results

Research questions

• RQ1 Does a general curve of genus g admit any exceptional secant planes, and if so, under what conditions?
• RQ2 How can the number of linear series with exceptional secant planes be computed in a one-parameter family using tautological classes?
• RQ3 What are the tautological coefficients of secant-plane formulas associated with linear series of degree 2d−1 and dimension m on curves of genus g?
• RQ4 Can a generating function be constructed for the number of d-secant (d−2)-planes on curves with exceptional linear series?
• RQ5 What is the formula for the number of exceptional linear series on a general curve that carries a one-dimensional family of such series?

Key findings

• A general curve of genus g has no exceptional secant planes, confirming a precise geometric non-existence result.
• The paper derives a formula for the number of linear series with exceptional secant planes on a general curve equipped with a one-dimensional family of such series.
• It proposes conjectural generating functions for the number of d-secant (d−2)-planes, expressed in terms of tautological coefficients.
• A strategy is developed to compute divisor classes in the Picard group of the moduli space of curves associated with exceptional secant plane behavior.
• The number of exceptional linear series in a one-parameter family is computed via tautological classes derived from the family's geometry.
• The framework successfully extends Brill-Noether theory to pairs of linear series, enabling the analysis of exceptional secant plane configurations.