Reducible deformations and smoothing of primitive multiple curves

TL;DR

This paper investigates the conditions under which primitive multiple curves can be deformed into smooth, reducible curves with transversal intersections, focusing on the role of certain line bundles and extending results to higher multiplicities.

Contribution

It establishes a criterion based on line bundle sections for deforming primitive double curves into smooth, reducible curves with transversal intersections, and explores properties for higher multiplicities.

Findings

Primitive double curves deform to smooth, reducible curves if and only if h^0(L^{-1}) ≠ 0.

Conditions for reducible deformations are characterized for multiplicity n > 2.

Properties of reducible deformations are analyzed for higher multiplicities.

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 $n>2$.