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Théorèmes de Petri pour les courbes stables et dégénérescence du système d'équation du plongement canonique

Abstract : Petri's theorem states that the canonical image of a nonhyperelliptic smooth curve of genus g>=4 defined over an algebraically closed field is an intersection of quadrics and cubics. Moreover, one can exhibit a system of equations for this image. These results are due to Petri (1923) and were generalized and transcribed in modern language by Saint-Donat (1973). The moduli space of smooth curves is not proper and can be completed by adding stable curves. It is therefore natural to search for generalizations of Petri's theorem for stable curves and to examine questions of degeneracy.

In this thesis, we consider on the one hand the case of a stable curve with one singular point and whose normalization is hyperelliptic, and on the other hand the case of a stable curve whose graph is planar. Moreover, we undertake the canonical embedding of a stable curve defined over a discrete valuation ring. The general method consists in:
-- describing the canonical sheaf and constructing a well adapted basis for the space of its global sections;
-- constructing quadrics and cubics in the canonical ideal;
-- proving that these equations generate the canonical ideal.

The text also contains new biographical indications concerning the german mathematician Karl Petri.
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Contributor : Olivier Dodane <>
Submitted on : Thursday, July 16, 2009 - 9:44:48 AM
Last modification on : Friday, June 19, 2020 - 9:22:04 AM
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  • HAL Id : tel-00392525, version 2



Olivier Dodane. Théorèmes de Petri pour les courbes stables et dégénérescence du système d'équation du plongement canonique. Mathématiques [math]. Université de Strasbourg, 2009. Français. ⟨NNT : 2009STRA6068⟩. ⟨tel-00392525v2⟩



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