Reducible deformations and smoothing of primitive multiple curves

Abstract

A primitive multiple curve is a Cohen-Macaulay irreducible projective curve $Y$ that can be locally embedded in a smooth surface, and such that $C=Y_{red}$ is smooth. In this case, $L={\mathcal{I}}_C/{\mathcal{I}}_C^2$ is a line bundle on $C$. This paper continues the study of deformations of $Y$ to curves with smooth irreducible components, when the number of components is maximal (it is then the multiplicity $n$ of $Y$). We prove that a primitive double curve can be deformed to reduced curves with smooth components intersecting transversally if and only if $h^0(L^{-1})\ne 0$. We give also some properties of reducible deformations in the case of multiplicity $n>2$.

22 pages

Cited by in corpus (1)

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