On Chow Stability for algebraic curves

TL;DR

This paper establishes a natural criterion for Chow and Hilbert stability of smooth projective curves based on the stability of the restricted tangent bundle, and describes the structure of the associated Hilbert scheme components.

Contribution

It introduces an intrinsic stability criterion for curves and characterizes a significant open subset of the Hilbert scheme of Chow-stable curves.

Findings

Stable tangent bundle implies Chow and Hilbert stability.

Describes an open subset of the Hilbert scheme of stable curves.

Provides a quotient stack description for these curves.

Abstract

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.