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arXiv · 2208.09745

On compactifications of $\mathcal{M}_{g,n}$ with colliding markings

Vance Blankers·Sebastian Bozlee
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Abstract

In this paper, we study all ways of constructing modular compactifications of the moduli space $\mathcal{M}_{g,n}$ of $n$-pointed smooth algebraic curves of genus $g$ by allowing markings to collide. We find that for any such compactification, collisions of markings are controlled by a simplicial complex which we call the collision complex. Conversely, we identify modular compactifications of $\mathcal{M}_{g,n}$ with essentially arbitrary collision complexes, including complexes not associated to any space of weighted pointed stable curves. These moduli spaces classify the modular compactifications of $\mathcal{M}_{g,n}$ by nodal curves with smooth markings as well as the modular compactifications of $\mathcal{M}_{1,n}$ with Gorenstein curves and smooth markings. These compactifications generalize previous constructions given by Hassett, Smyth, and Bozlee--Kuo--Neff.

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Vance Blankers, Sebastian Bozlee. 2022-08-20. On compactifications of $\mathcal{M}_{g,n}$ with colliding markings. https://arxiv.org/abs/2208.09745

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