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Extremal effective divisors of Brill-Noether and Gieseker-Petri type in $$\overline{\mathcal{M}}_{1,n}$$. (English) Zbl 1359.14026
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 $$f^*\widetilde{BN^1_3}$$.
##### 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
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