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Affine geometry of strata of differentials. (English) Zbl 1423.14184
Dans cet article, l’auteur étudie le problème constituant à savoir si les strates de différentielles abéliennes sont des variétés affines (au sens de la géométrie algébrique). L’auteur prouve que les composantes hyperelliptiques des strates \(\Omega\mathcal{M}_g(2g-2)\) et \(\Omega\mathcal{M}_g(g-1,g-1)\), ainsi que quelques strates de petites dimensions sont affines. De plus il est montré que les courbes de Teichmüller et les espaces d’Hurwitz de revêtements du tore sont affines. La question reste ouverte pour les autres strates, toutefois l’auteur montre que les strates de différentielles méromorphes ne possèdent pas de courbes complètes.
Les preuves font essentiellement intervenir des notions de géométrie algébrique classique. Combiné avec une rédaction soignée, cela rend cet article accessible à un large public.

MSC:
14H10 Families, moduli of curves (algebraic)
14H15 Families, moduli of curves (analytic)
32G15 Moduli of Riemann surfaces, Teichmüller theory (complex-analytic aspects in several variables)
14H55 Riemann surfaces; Weierstrass points; gap sequences
30F60 Teichmüller theory for Riemann surfaces
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