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arXiv · 1904.03465

R\'eduction stable en dimension sup\'erieure [d'apr\`es Koll\'ar, Hacon-Xu...]

Olivier Benoist
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Abstract

The moduli space of stable curves of Deligne and Mumford is a compactification of the moduli space of smooth curves of genus >=2 that parametrizes certain nodal curves. It is a powerful tool for the study of algebraic curves. Higher-dimensional analogues were constructed by Koll\'ar, Shepherd-Barron and Alexeev in dimension 2, and by Viehweg in the case of smooth varieties. We will explain the recent ideas allowing for the construction of these moduli spaces in general, including the stable reduction theorem in higher dimension, which reflects their compactness. L'espace de modules des courbes stables de Deligne et Mumford est une compactification de l'espace de modules des courbes lisses de genre >=2, param\'etrant certaines courbes nodales. C'est un outil puissant pour l'\'etude des courbes alg\'ebriques. Des analogues en dimension sup\'erieure ont \'et\'e construits par Koll\'ar, Shepherd-Barron et Alexeev en dimension 2, et par Viehweg dans le cas des vari\'et\'es lisses. Nous expliquerons les id\'ees r\'ecentes ayant permis la construction de ces espaces de modules en g\'en\'eral, notamment le th\'eor\`eme de r\'eduction stable en dimension sup\'erieure, qui refl\`ete leur compacit\'e.

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Olivier Benoist. 2019-04-06. R\'eduction stable en dimension sup\'erieure [d'apr\`es Koll\'ar, Hacon-Xu...]. https://doi.org/10.24033/ast.1137

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