Schematic Harder Narasimhan stratification

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.