The dual complex of ${\mathcal{M}}_{1,n}({\mathbb{P}}^r,d)$ via the geometry of the Vakil--Zinger moduli space

Abstract

We study normal crossings compactifications of the moduli space of maps ${\mathcal{M}}_{g,n}({\mathbb{P}}^r,d)$, for $g=0$ and $g=1$. In each case we explicitly determine the dual boundary complex, and prove that it admits a natural interpretation as a moduli space of decorated metric graphs. We prove that the dual complexes are contractible when $r\ge 1$ and $d>g$. When $g=1$, our result depends on a new understanding of the connected components of boundary strata in the Vakil--Zinger desingularization and its modular interpretation by Ranganathan--Santos-Parker--Wise.

arXiv commentv2: expanded the proofs of strata irreducibility, improved exposition, accepted at Adv. Math

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