On Chow stability for algebraic curves

In the last decades there have been introduced different concepts of stability for projective varieties. In this paper we give a natural and intrinsic criterion for the Chow and Hilbert stability for complex irreducible smooth projective curves . Namely, if the restriction of the tangent bundle of to C is stable then is Chow stable, and hence Hilbert stable. We apply this criterion to describe a smooth open set of a regular component of the locus of Chow stable curves of the Hilbert scheme of with Hilbert polynomial , when and Moreover, we describe the quotient stack of such curves. Similar results are obtained for the locus of Hilbert stable curves.