Blowing-ups of primitive multiple schemes

Abstract

A primitive multiple scheme is a Cohen-Macaulay scheme $\bf X$ such that the associated reduced scheme $X={\bf X}_{red}$ is smooth, irreducible, and that $\bf X$ can be locally embedded in a smooth variety of dimension $\dim(X)+1$. If ${\bf I}_X$ is the ideal sheaf of $X$, $L={\bf I}_X/{\bf I}_X^2$ is a line bundle on $X$, called the associated line bundle of $\bf X$. The simplest example is the trivial primitive multiple scheme of multiplicity $n$ associated to a line bundle $L$ on $X$: it is the $n$-th infinitesimal neighborhood of $X$, embedded in the line bundle $L^*$ by the zero section. A subscheme $Z$ of ${\bf X}$ is called good if $Z_{red}$ is smooth and connected, and if $d={\rm codim}(Z)$ and ${\bf I}_{Z,{\bf X}}$ is the ideal sheaf of $Z$, then for every closed point $z\in Z$, ${\bf I}_{Z,{\bf X},z}$ can be generated by $d$ elements. Two kinds of subschemes $Z$ of ${\bf X}$ will be considered: the closed smooth subschemes of $X$, seen as subschemes of $\bf X$, and the good subschemes. In the two cases, the blowing-up ${\bf B}_{Z,{\bf X}}$ of ${\bf X}$ along $Z$ is a primitive multiple scheme of multiplicity $n$, and its underlying smooth scheme is the blowing-up ${\bf B}_{Z_{red},X}$ of $X$ along $Z_{red}$. Additional results are obtained in the case of hypersurfaces or points of $X$. We treat the case of $X={\rm P}_2$, with $Z$ a single point $P$. Let $p:\widetilde{{\rm P}}_2\to{\rm P}_2$ be the blowing-up of ${\rm P}_2$ along $P$. We find all primitive double schemes $\bf X$, $\bf Y$, with ${\bf Y}_{red}=\widetilde{{\rm P}}_2$, ${\bf X}_{red}={\rm P}_2$, such that there is a morphism ${\rm Y}\to{\rm X}$ inducing $p$ and an isomorphism ${\bf Y}\backslash p^{-1}(P)\to{\bf X} \backslash\{P\}$. We obtain in this way the list of all K3-carpets with underlying smooth variety $\widetilde{{\rm P}}_2$.