Effective divisors on $${\overline{\mathcal{M}}_g}$$ associated to curves with exceptional secant planes

Abstract

This article is a sequel to Cotterill (Math Zeit 267(3):549–582, 2011), in which the author studies secant planes to linear series on a curve that is general in moduli. In that paper, the author proves that a general curve has no linear series with exceptional secant planes, in a very precise sense. Consequently, it makes sense to study effective divisors on \({\overline{\mathcal{M}}_g}\) associated to curves equipped with secant-exceptional linear series. Here we describe a strategy for computing the classes of those divisors. We pay special attention to the extremal case of (2d − 1)-dimensional series with d-secant (d − 2)-planes, which appears in the study of Hilbert schemes of points on surfaces. In that case, modulo a combinatorial conjecture, we obtain hypergeometric expressions for tautological coefficients that enable us to deduce the asymptotics in d of our divisors’ virtual slopes.