The Deligne–Mumford and the Incidence Variety Compactifications of the Strata of Ω g
[Les compactifications de Deligne–Mumford et de la variété d’incidence des strates de Ω g ]
Annales de l'Institut Fourier, Tome 68 (2018) no. 3, pp. 1169-1240.

L’objectif central de cet article est de construire et d’étudier une compactification raisonnable des strates des différentielles abéliennes. L’ingrédient principal pour l’étude de cette compactification des strates est une généralisation des techniques de plomberie cylindrique aux différentielles. Cette compactification nous permet de calculer la dimension de Kodaira de certaines de ces strates. Un autre résultat digne d’intérêt est le calcul de la dimension de la projection des strates dans l’espace des modules des surfaces de Riemann. Enfin nous étudions certains problèmes liés à la parité des strates au bord, les composantes hyperelliptiques ainsi que la strate minimale en genre trois.

The main goal of this work is to construct and study a reasonable compactification of the strata of the moduli space of abelian differentials. This allows us to compute the Kodaira dimension of some strata of the moduli space of abelian differentials. The main ingredients to study the compactifications of the strata are a version of the plumbing cylinder construction for differential forms and an extension of the parity of the connected components of the strata to the differentials on curves of compact type. We study in detail the compactifications of the hyperelliptic minimal strata and of the odd minimal stratum in genus three.

Reçu le :
Révisé le :
Accepté le :
Publié le :
DOI : 10.5802/aif.3187
Classification : 14H15, 30F30, 14E99, 14H45
Keywords: Abelian differentials, Riemann surfaces, Moduli spaces, Strata, Compactification, Kodaira dimension
Mot clés : Différentielles abéliennes, Surfaces de Riemann, Espace des modules, Strates, Compactifications, Dimension de Kodaira
Gendron, Quentin 1

1 Institut für Mathematik, Goethe-Universität, Robert-Mayer-Str. 6-8, D-60325 Frankfurt am Main (Germany)
@article{AIF_2018__68_3_1169_0,
     author = {Gendron, Quentin},
     title = {The {Deligne{\textendash}Mumford} and the {Incidence} {Variety} {Compactifications} of the {Strata} of $\Omega \protect \mathcal{M}_{g}$},
     journal = {Annales de l'Institut Fourier},
     pages = {1169--1240},
     publisher = {Association des Annales de l{\textquoteright}institut Fourier},
     volume = {68},
     number = {3},
     year = {2018},
     doi = {10.5802/aif.3187},
     language = {en},
     url = {http://archive.numdam.org/articles/10.5802/aif.3187/}
}
TY  - JOUR
AU  - Gendron, Quentin
TI  - The Deligne–Mumford and the Incidence Variety Compactifications of the Strata of $\Omega \protect \mathcal{M}_{g}$
JO  - Annales de l'Institut Fourier
PY  - 2018
SP  - 1169
EP  - 1240
VL  - 68
IS  - 3
PB  - Association des Annales de l’institut Fourier
UR  - http://archive.numdam.org/articles/10.5802/aif.3187/
DO  - 10.5802/aif.3187
LA  - en
ID  - AIF_2018__68_3_1169_0
ER  - 
%0 Journal Article
%A Gendron, Quentin
%T The Deligne–Mumford and the Incidence Variety Compactifications of the Strata of $\Omega \protect \mathcal{M}_{g}$
%J Annales de l'Institut Fourier
%D 2018
%P 1169-1240
%V 68
%N 3
%I Association des Annales de l’institut Fourier
%U http://archive.numdam.org/articles/10.5802/aif.3187/
%R 10.5802/aif.3187
%G en
%F AIF_2018__68_3_1169_0
Gendron, Quentin. The Deligne–Mumford and the Incidence Variety Compactifications of the Strata of $\Omega \protect \mathcal{M}_{g}$. Annales de l'Institut Fourier, Tome 68 (2018) no. 3, pp. 1169-1240. doi : 10.5802/aif.3187. http://archive.numdam.org/articles/10.5802/aif.3187/

[1] Arbarello, Enrico; Cornalba, Maurizio; Griffiths, Phillip A. Geometry of algebraic curves. Volume II, Grundlehren der Mathematischen Wissenschaften, 268, Springer, Heidelberg, 2011, xxx+963 pages | MR | Zbl

[2] Aulicino, David; Nguyen, Duc-Manh; Wright, Alex Classification of higher rank orbit closures in odd (4)., J. Eur. Math. Soc. (JEMS), Volume 18 (2016) no. 8, pp. 1855-1872 | DOI | Zbl

[3] Bainbridge, Matt; Chen, Dawei; Gendron, Quentin; Grushevsky, Samuel; Moeller, Martin Compactification of strata of abelian differentials (to appear in Duke Math. J.)

[4] Bardelli, Fabio Lectures on stable curves, Lectures on Riemann surfaces (Trieste, 1987), World Scientific Publishing Co, 1989, pp. 648-704 | Zbl

[5] Teixidor i Bigas, Montserrat The divisor of curves with a vanishing theta-null, Compos. Math., Volume 66 (1988) no. 1, pp. 15-22 | MR | Zbl

[6] Bini, Gilberto; Fontanari, Claudio; Viviani, Filippo On the birational geometry of the universal Picard variety., Int. Math. Res. Not., Volume 2012 (2012) no. 4, pp. 740-780 | DOI | Zbl

[7] Chen, Dawei Covers of elliptic curves and the moduli space of stable curves, J. Reine Angew. Math., Volume 649 (2010), pp. 167-205 | DOI | MR | Zbl

[8] Chen, Dawei Degenerations of Abelian Differentials, J. Differ. Geom., Volume 107 (2017) no. 3, pp. 395-453 | DOI | Zbl

[9] Cornalba, Maurizio Moduli of curves and theta-characteristics, Lectures on Riemann surfaces (Trieste, 1987), World Scientific Publishing Co, 1989, pp. 560-589 | MR | Zbl

[10] Deligne, Pierre; Mumford, David The irreducibility of the space of curves of given genus, Publ. Math., Inst. Hautes Étud. Sci. (1969) no. 36, pp. 75-109 | DOI | MR | Zbl

[11] Eskin, Alex; Masur, Howard; Zorich, Anton Moduli spaces of abelian differentials: the principal boundary, counting problems, and the Siegel-Veech constants, Publ. Math., Inst. Hautes Étud. Sci. (2003) no. 97, pp. 61-179 | DOI | MR | Zbl

[12] Farkas, Gavril The birational type of the moduli space of even spin curves., Adv. Math., Volume 223 (2010) no. 2, pp. 433-443 | DOI | Zbl

[13] Farkas, Gavril; Pandharipande, Rahul The moduli space of twisted canonical divisors (to appear in J. Inst. Math. Jussieu)

[14] Farkas, Gavril; Verra, Alessandro Moduli of theta-characteristics via Nikulin surfaces., Math. Ann., Volume 354 (2012) no. 2, pp. 465-496 | DOI | Zbl

[15] Farkas, Gavril; Verra, Alessandro The geometry of the moduli space of odd spin curves., Ann. Math., Volume 180 (2014) no. 3, pp. 927-970 | DOI | Zbl

[16] Griffiths, Phillip A.; Harris, Joseph Principles of algebraic geometry, Wiley Classics Library, John Wiley & Sons, New York, 1994, xiv+813 pages (Reprint of the 1978 original) | MR | Zbl

[17] Harris, Joe; Morrison, Ian Moduli of curves, Graduate Texts in Mathematics, 187, Springer, New York, 1998, xiv+366 pages | MR | Zbl

[18] Harris, Joe; Mumford, David On the Kodaira dimension of the moduli space of curves, Invent. Math., Volume 67 (1982) no. 1, pp. 23-88 (With an appendix by William Fulton) | DOI | MR | Zbl

[19] Johnson, Dennis Spin structures and quadratic forms on surfaces, J. Lond. Math. Soc., Volume 22 (1980) no. 2, pp. 365-373 | DOI | MR | Zbl

[20] Kang, Pyung-Lyun On singular plane quartics as limits of smooth curves of genus three, J. Korean Math. Soc., Volume 37 (2000) no. 3, pp. 411-436 | MR | Zbl

[21] Kontsevich, Maxim; Zorich, Anton Connected components of the moduli spaces of Abelian differentials with prescribed singularities, Invent. Math., Volume 153 (2003) no. 3, pp. 631-678 | DOI | MR | Zbl

[22] Matheus, Carlos; Möller, Martin; Yoccoz, Jean-Christophe A criterion for the simplicity of the Lyapunov spectrum of square-tiled surfaces., Invent. Math., Volume 202 (2015) no. 1, pp. 333-425 | DOI | Zbl

[23] Mumford, David Curves and their Jacobians., The University of Michigan Press, 1975, 104 pages | Zbl

[24] de Saint-Gervais, Henri Paul Uniformisation des surfaces de Riemann, ENS Éditions, 2010, 544 pages | MR | Zbl

[25] Ueno, Kenji Classification theory of algebraic varieties and compact complex spaces, Lecture Notes in Mathematics, 439, Springer, 1975, xix+278 pages (Notes written in collaboration with P. Cherenack) | MR | Zbl

[26] Viehweg, Eckart Die Additivität der Kodaira Dimension für projektive Faserräume über Varietäten des allgemeinen Typs, J. Reine Angew. Math., Volume 330 (1982), pp. 132-142 | DOI | MR | Zbl

[27] Zorich, Anton Flat surfaces, Frontiers in number theory, physics, and geometry. I, Springer, Berlin, 2006, pp. 437-583 | DOI | MR | Zbl

Cité par Sources :