Remarks about bubbles
Abstract
We make some remarks about bubbling on, not necessarily proper, champs de Deligne-Mumford, i.e. compactification of the space of mappings from a given (wholly scheme like) curve, so, in particular, on quasi-projective projective varieties. Under hypothesis on both the interior and the boundary such as Remark rmk:interior below, this implies an optimal logarithmic variant of Mori's Bend-and-Break. The main technical remark is thm:logMM, while our final remark, the cone theorem, rmk:cone, is a variant.
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