Abstract

The Harder Narasimhan type (in the sense of Gieseker semistability) of a pure-dimensional coherent sheaf on a projective scheme is known to vary semi-continuously in a flat family, which gives the well-known Harder Narasimhan stratification of the parameter scheme of the family, by locally closed subsets. We show that each stratum can be endowed with a natural structure of a locally closed subscheme of the parameter scheme, which enjoys an appropriate universal property.

As an application, we deduce that pure-dimensional coherent sheaves of any given Harder Narasimhan type form an Artin algebraic stack.

As another application – jointly with L. Brambila-Paz and O. Mata – we describe moduli schemes for certain rank 2 unstable vector bundles on a smooth projective curve, fixing some numerical data.