Pré-Publication, Document De Travail Année : 2025

Gluing fat points and singular algebraic curves

Recollement le long de points épais et points singuliers de courbes

José Bertin
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Résumé

We study singular curves, algebroid or projective, obtained by gluing two fat points on smooth curves, generalizing a wellknown construction of a singular curve from a smooth curve and a module on it. We study this class of singularities both from a local and global point of view. These are the singular curves such that the normalization morphism is a strict epimorphism. Plane branches of multiplicity greater than 2 cannot be obtained by gluing. Every algebroid curve S has a closure S ⊂ S which is the smallest algebroid curve containing S obtained by gluing two fat points. We will see that S and S are equisingular in Zariski's sense. To do this we will compare S with the saturation of S.

A key role is played by an algebra introduced by Tessier. This algebra not only contains a great deal of informations about S, but also provides a relationship between the equisingular moduli space of planar branches and the punctual Hilbert scheme of the plane. We emphasize the usefulness of this algebra in both settings, plane branches and monomial branches.

In conclusion, we show how to describe all vector bundles on these singular curves by gluing together free modules along the fat points, a variant of the so-called "sandwich" construction of Drozd, Greuel and others.

On étudie les points singuliers de branches et de courbes projectives obtenues par recollement le long de deux points épaissis. Un rôle important est joué par une algèbre introduite par B. Tessier. Comparaison est faite avec l'opération de saturation de Zariski.

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hal-04979214 , version 1 (09-03-2025)

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José Bertin. Gluing fat points and singular algebraic curves. 2025. ⟨hal-04979214⟩
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