Covers of stacky curves and limits of plane quintics
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Abstract:
We construct a well-behaved compactification of the space of finite covers of a stacky curve using admissible cover degenerations. Using our construction, we compactify the space of tetragonal curves on Hirzebruch surfaces. As an application, we explicitly describe the boundary divisors of the closure in $\overline {\mathcal M}_6$ of the locus of smooth plane quintic curves.References
- Dan Abramovich, Alessio Corti, and Angelo Vistoli, Twisted bundles and admissible covers, Comm. Algebra 31 (2003), no. 8, 3547–3618. Special issue in honor of Steven L. Kleiman. MR 2007376, DOI 10.1081/AGB-120022434
- Dan Abramovich and Angelo Vistoli, Compactifying the space of stable maps, J. Amer. Math. Soc. 15 (2002), no. 1, 27–75. MR 1862797, DOI 10.1090/S0894-0347-01-00380-0
- Valery Alexeev, Moduli spaces $M_{g,n}(W)$ for surfaces, Higher-dimensional complex varieties (Trento, 1994) de Gruyter, Berlin, 1996, pp. 1–22. MR 1463171, DOI 10.1006/jcat.1996.0357
- Alessandro Chiodo, Stable twisted curves and their $r$-spin structures, Ann. Inst. Fourier (Grenoble) 58 (2008), no. 5, 1635–1689 (English, with English and French summaries). MR 2445829
- Charles F. Doran, Andrew Harder, Andrey Y. Novoseltsev, and Alan Thompson, Families of lattice polarized K3 surfaces with monodromy, Int. Math. Res. Not. IMRN 23 (2015), 12265–12318. MR 3431621, DOI 10.1093/imrn/rnv071
- William Fulton and Robert MacPherson, A compactification of configuration spaces, Ann. of Math. (2) 139 (1994), no. 1, 183–225. MR 1259368, DOI 10.2307/2946631
- Edmond E. Griffin II, Families of quintic surfaces and curves, Compositio Math. 55 (1985), no. 1, 33–62. MR 791646
- Paul Hacking, Compact moduli of plane curves, Duke Math. J. 124 (2004), no. 2, 213–257. MR 2078368, DOI 10.1215/S0012-7094-04-12421-2
- Joe Harris, Theta-characteristics on algebraic curves, Trans. Amer. Math. Soc. 271 (1982), no. 2, 611–638. MR 654853, DOI 10.1090/S0002-9947-1982-0654853-6
- Joe Harris and David Mumford, On the Kodaira dimension of the moduli space of curves, Invent. Math. 67 (1982), no. 1, 23–88. With an appendix by William Fulton. MR 664324, DOI 10.1007/BF01393371
- Brendan Hassett, Stable log surfaces and limits of quartic plane curves, Manuscripta Math. 100 (1999), no. 4, 469–487. MR 1734796, DOI 10.1007/s002290050213
- Brendan Hassett, Local stable reduction of plane curve singularities, J. Reine Angew. Math. 520 (2000), 169–194. MR 1748273, DOI 10.1515/crll.2000.020
- G. Horrocks, Vector bundles on the punctured spectrum of a local ring, Proc. London Math. Soc. (3) 14 (1964), 689–713. MR 169877, DOI 10.1112/plms/s3-14.4.689
- J. Kollár and N. I. Shepherd-Barron, Threefolds and deformations of surface singularities, Invent. Math. 91 (1988), no. 2, 299–338. MR 922803, DOI 10.1007/BF01389370
- Johan Martens and Michael Thaddeus, Variations on a theme of Grothendieck, arXiv:1210.8161 [math.AG] (2012).
- Martin C. Olsson, (Log) twisted curves, Compos. Math. 143 (2007), no. 2, 476–494. MR 2309994, DOI 10.1112/S0010437X06002442
- Flavia Poma, Étale cohomology of a DM curve-stack with coefficients in $\Bbb G_m$, Monatsh. Math. 169 (2013), no. 1, 33–50. MR 3016518, DOI 10.1007/s00605-012-0450-5
- Sevin Recillas, Maps between Hurwitz spaces, Bol. Soc. Mat. Mexicana (2) 18 (1973), 59–63. MR 360603
- David Rydh, The canonical embedding of an unramified morphism in an étale morphism, Math. Z. 268 (2011), no. 3-4, 707–723. MR 2818725, DOI 10.1007/s00209-010-0691-8
- Ravi Vakil, Twelve points on the projective line, branched covers, and rational elliptic fibrations, Math. Ann. 320 (2001), no. 1, 33–54. MR 1835061, DOI 10.1007/PL00004469
Additional Information
- Anand Deopurkar
- Affiliation: Department of Mathematics, Boyd GSRC, University of Georgia, Athens, Georgia, 30605
- MR Author ID: 1037149
- ORCID: 0000-0002-3250-2342
- Email: anand.deopurkar@anu.edu.au
- Received by editor(s): August 23, 2016
- Received by editor(s) in revised form: April 26, 2017
- Published electronically: July 12, 2018
- © Copyright 2018 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 371 (2019), 549-588
- MSC (2010): Primary 14H10, 14H45, 14H50
- DOI: https://doi.org/10.1090/tran/7301
- MathSciNet review: 3885154