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Varieties of clusters and Enriques diagrams

Published online by Cambridge University Press:  07 July 2004

JOAQUIM ROÉ
Affiliation:
Departament de Matemàtiques, Universitat Autônoma de Barcelona, 08193, Bellaterra, Spain. e-mail: [email protected]

Abstract

Given a surface $S$ and an integer $r\,{\ge}\,1$, there is a variety $X_{r-1}$ parametrizing all clusters of $r$ proper and infinitely near points of $S$. We study the geometry of the varieties $X_r$, showing that for every Enriques diagram ${\bf D}$ of $r$ vertices the subset ${\mathit{Cl}}({\bf D})\,{\subset}\,X_{r-1}$ of the clusters with Enriques diagram ${\bf D}$ is locally closed. We study also the relative positions of the subvarieties ${\mathit{Cl}}({\bf D})$, showing that they do not form a stratification and giving criteria for adjacencies between them.

Type
Research Article
Copyright
2004 Cambridge Philosophical Society

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