Extremal effective divisors of Brill-Noether and Gieseker-Petri type in \(\overline{\mathcal{M}}_{1,n}\). (English) Zbl 1359.14026

Chen, Dawei; Patel, Anand

Adv. Geom. 16, No. 2, 231-242 (2016).

The goal of the paper is to show the existence of extremal effective divisors on the moduli space \(\overline{M}_{1,n}\) of genus one curves with \(n\geq 6\) marked points, which are not of the type \[ \overline{\{(E,p_1,\ldots, p_n) \;| \;\sum_i a_i p_i \sim 0 \;\text{in } E \}} \] for coprime integers \(a_i\) summing to \(0\).
For \(n\geq 8\), the authors construct them as pull-backs of some Brill-Noether divisors over \(\overline{M}_g\) for suitable \(g\). More precisely, they perform a construction using trigonal curves that works for any \(n\geq 8\), and \(d\)-gonal curves for \(n=4d-4 \geq 12\).
For \(n\geq 6\), the authors construct another class of such divisors using the Gieseker-Petri divisor in \(\overline{M}_4\) (that is, the divisor of curves whose canonical image in \(\mathbb{P}^3\) is contained in a quadric cone).
The proofs are explicit and the paper is essentially self-contained.
Reviewer’s remark: In Theorem 3.3, the 8 in \(f^*\widetilde{BN^1_8}\) appears to be a typo; it should be replaced by \(f^*\widetilde{BN^1_3}\).

Reviewer: Simone Melchiorre Chiarello (Onex)

MSC:

14H10	Families, moduli of curves (algebraic)
14H45	Special algebraic curves and curves of low genus
14H51	Special divisors on curves (gonality, Brill-Noether theory)
14Cxx	Cycles and subschemes

Keywords:

moduli space of curves of genus 1; extremal divisors