Reducible deformations and smoothing of primitive multiple curves
Jean-Marc Drézet
TL;DR
This work addresses when primitive multiple curves $Y=C_n$ admit maximal reducible deformations to $n$ smooth components. It establishes a cohomological criterion for primitive double curves: a local reducible deformation exists if and only if $h^0(L^*)>0$, and a global maximal deformation exists when $L\simeq \mathcal{O}_C(-P_1-\cdots-P_p)$; these results are extended to general multiplicity $n\ge 2$ by introducing a spectrum $ (p_{ij}) $ controlling component intersections and constructing deformations via cocycle gluings. Consequently, if $L$ has the form $\mathcal{O}_C(-\Delta)$ for an effective divisor $\Delta$, a global maximal reducible deformation exists, implying the smoothability of the primitive multiple curve. The paper also discusses implications for the moduli of coherent sheaves on such curves and frames a path from local to global deformation theory through explicit cocycle methods.
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 $h^0(L^{-1})\not=0$. We give also some properties of reducible deformations in the case of multiplicity $n>2$.