Quadratic differentials in low genus: exceptional and non-varying strata
[Différentielles quadratiques en petit genre : strates exceptionnelles et strates sans variance]
Annales scientifiques de l'École Normale Supérieure, Série 4, Tome 47 (2014) no. 2, pp. 309-369.

Nous présentons une façon algébrique de distinguer les composantes exceptionnelles des strates de l'espace de modules des différentielles quadratiques en genres trois et quatre. La liste complète de ces strates est (9,-1), (6,3,-1) et (3,3,3,-1) en genre trois, (12), (9,3), (6,6), (6,3,3) et (3,3,3,3) en genre quatre, respectivement. La distinction est basée sur des propriétés géométriques du modèle canonique de ces courbes.

Ce résultat fait partie de la détermination de la somme des exposants de Lyapunov des courbes de Teichmüller, dans la continuité de [9], [6] et [7]. Pour beaucoup de strates en petit genre les courbes de Teichmüller sont disjointes des diviseurs de type Brill-Noether. On en déduit que la somme des exposants de Lyapunov de toute courbe de Teichmüller dans ces strates est égale à la somme des exposants pour la mesure à support sur toute la strate.

We give an algebraic way of distinguishing the components of the exceptional strata of quadratic differentials in genus three and four. The complete list of these strata is (9,-1), (6,3,-1), (3,3,3,-1) in genus three and (12), (9,3), (6,6), (6,3,3) and (3,3,3,3) in genus four. The upshot of our method is a detailed study regarding the geometry of canonical curves.

This result is part of a more general investigation about the sum of Lyapunov exponents of Teichmüller curves, building on [9], [6] and [7]. Using disjointness of Teichmüller curves with divisors of Brill-Noether type on the moduli space of curves, we show that for many strata of quadratic differentials in low genus the sum of Lyapunov exponents for the Teichmüller geodesic flow is the same for all Teichmüller curves in that stratum.

Publié le :
DOI : 10.24033/asens.2216
Classification : 14H10; 14H51, 37D40
Keywords: Teichmüller curve, sum of Lyapunov exponents, canonical model, Brill-Noether divisor, exceptional strata.
Mot clés : Courbe de Teichmüller, somme des exposants de Lyapunov, modèle canonique, diviseur de Brill-Noether, strates exceptionnelles. 
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     author = {Chen, Dawei and M\"oller, Martin},
     title = {Quadratic differentials in low genus: exceptional and non-varying strata},
     journal = {Annales scientifiques de l'\'Ecole Normale Sup\'erieure},
     pages = {309--369},
     publisher = {Soci\'et\'e Math\'ematique de France. Tous droits r\'eserv\'es},
     volume = {Ser. 4, 47},
     number = {2},
     year = {2014},
     doi = {10.24033/asens.2216},
     mrnumber = {3215925},
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Chen, Dawei; Möller, Martin. Quadratic differentials in low genus: exceptional and non-varying strata. Annales scientifiques de l'École Normale Supérieure, Série 4, Tome 47 (2014) no. 2, pp. 309-369. doi : 10.24033/asens.2216. http://www.numdam.org/articles/10.24033/asens.2216/

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