On Chow Stability for algebraic 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 $Hil{b}_{Ch}^{P(t),s}$ of the Hilbert scheme of ${\mathbb{P}}^n$ containing the generic smooth Chow-stable curve of genus $g$ and degree $d>g+n-\lfloor \frac{g}{n+1}\rfloor .$ Moreover, we describe the quotient stack of such curves. Similar results are obtained for the locus of Hilbert stable curves.

Minor corrections and improvements to presentation. We add Theorem 4.2

References in corpus (5)

• Moduli of varieties of general type
• K-polystability of Q-Fano varieties admitting Kahler-Einstein metrics
• Alternate Compactifications of Moduli Spaces of Curves
• GIT for polarized curves
• GIT Constructions of Log Canonical Models of M_g

Cited by in corpus (1)

• Limit linear series moduli stacks in higher rank