On the singular sheaves in the fine Simpson moduli spaces of $1$-dimensional sheaves

Abstract

In the Simpson moduli space $M$ of semi-stable sheaves with Hilbert polynomial $dm-1$ on a projective plane we study the closed subvariety $M^{\prime }$ of sheaves that are not locally free on their support. We show that for $d\ge 4$ it is a singular subvariety of codimension $2$ in $M$. The blow up of $M$ along $M^{\prime }$ is interpreted as a (partial) modification of $M\setminus M^{\prime }$ by vector bundles (on support).

arXiv commentDetails added, typos fixed; a new section on modifications added