Quasi locally free sheaves on primitive double schemes

Abstract

Let $X_2$ be a complex primitive double scheme, $X=(X_2)_{red}$. The ideal sheaf of $X$ is a line bundle on $X$. Let $E$ be a coherent sheaf on $X_2$. Then there exists a nonempty open subset $U$ of $X_2$ and integers $m_1$, $m_2$, uniquely determined by $E$, such that $E_{|U}$ is locally isomorphic to $(O_X\otimes C^{m_1})\oplus(O_{X_2}\otimes C^{m_2})$. If we can take $U=X_2$, we say the $E$ is quasi locally free. If $m_1=1$ we say that $E$ is rigid. In this case, we show that deformations of $E$ are of the same type. So locally free sheaves and rigid quasi locally free sheaves on $X_2$ play the same role on $X_2$ as locally free sheaves do on smooth schemes. In this paper we study the existence and description of quasi locally free sheaves, and flat families of such sheaves.