On the locus of genus 3 curves that admit meromorphic differentials with a zero of order 6 and a pole of order 2
[Sur le lieu défini par les courbes de genre 3 qui admettent une différentielle méromorphe avec un zéro d’ordre 6 et un pôle d’ordre 2.]
Annales de l'Institut Fourier, Tome 72 (2022) no. 1, pp. 261-299.

Le but principal de cet article est de calculer la classe du diviseur de ¯ 3 qu’on obtient en prenant la fermeture de l’image de la strate Ω 3 (6;-2) par la fonction qui oublie la différentielle. Ceci est réalisé via la formule de Porteous et la théorie des courbes test. À cette fin, nous étudions certaines propriétés du lieu des différentielles méromorphes de seconde espèce, i.e. dont tous les résidus sont nuls. Nous calculons la dimension de l’image de ce lieu dans g par l’application d’oubli pour tout g et pour toute partition. De plus, nous résolvons certains problèmes énumératifs impliquant ces différentielles en petit genre. L’outil clef de la preuve est la compactification des strates de différentielles abéliennes récemment introduite par Bainbridge–Chen–Gendron–Grushevsky–Möller.

The main goal of this article is to compute the class of the divisor of ¯ 3 obtained by taking the closure of the image of Ω 3 (6;-2) by the forgetful map. This is done using Porteous formula and the theory of test curves. For this purpose, we study the locus of meromorphic differentials of the second kind, computing the dimension of the map of these loci to g and solving some enumerative problems involving such differentials in low genus. A key tool of the proof is the compactification of strata recently introduced by Bainbridge–Chen–Gendron–Grushevsky–Möller.

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DOI : 10.5802/aif.3472
Classification : 30F30, 14H10, 32G15, 14H45
Keywords: Abelian differentials, Differentials of second kind, Moduli Space of curves, Deligne–Mumford compactification, Picard group
Mot clés : Différentielles abéliennes, Différentielles de seconde espèce, Espace des modules des courbes, Compactification de Deligne–Mumford compactification, Groupe de Picard
Castorena, Abel 1 ; Gendron, Quentin 2, 3

1 Centro de Ciencias Matemáticas-UNAM, Antigua Car. a Pátzcuaro 8701, Col. Ex Hacienda San José de la Huerta, Morelia, Mich., México
2 Centro de Ciencias Matemáticas-UNAM, Antigua Car. a Pátzcuaro 8701, Col. Ex Hacienda San José de la Huerta, Morelia, Mich. (México)
3 Centro de Investigación en Matemáticas, Guanjuato, Gto., AP 402, CP 36000 (México)
Licence : CC-BY-ND 4.0
Droits d'auteur : Les auteurs conservent leurs droits
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Castorena, Abel; Gendron, Quentin. On the locus of genus 3 curves that admit meromorphic differentials with a zero of order 6 and a pole of order 2. Annales de l'Institut Fourier, Tome 72 (2022) no. 1, pp. 261-299. doi : 10.5802/aif.3472. https://aif.centre-mersenne.org/articles/10.5802/aif.3472/

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