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 of the Chow, and Hilbert, stability for complex irreducible smooth projective curves $C\subset \mathbb P ^n$. Namely, if the restriction $T\mathbb P_{|C} ^n$ of the tangent bundle of $\mathbb P ^n$ to $C$ is stable then $C\subset \mathbb P ^n$ is Chow stable, and hence Hilbert stable. We apply this criterion to describe a smooth open set of the irreducible component $Hilb^{P(t),s}_{{Ch}}$ of the Hilbert scheme of $\mathbb{P} ^n$ containing the generic smooth Chow-stable curve of genus $g$ and degree $d>g+n-\left\lfloor\frac{g}{n+1}\right\rfloor.$ Moreover, we describe the quotient stack of such curves. Similar results are obtained for the locus of Hilbert stable curves.