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, Volume 151, Issue 3–4, pp 289–304 | Cite as

On Chow stability for algebraic curves

  • L. Brambila-PazEmail author
  • H. Torres-López


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 \(C\subset {\mathbb {P}} ^n\). Namely, if the restriction 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 a regular component of the locus of Chow stable curves of the Hilbert scheme of \(\mathbb {P} ^n\) with Hilbert polynomial \(P(t)=dt+(1-g)\), when \(g\ge 4\) and \(d>g+n-\left\lfloor \frac{g}{n+1}\right\rfloor .\) Moreover, we describe the quotient stack of such curves. Similar results are obtained for the locus of Hilbert stable curves.

Mathematics Subject Classification

14H60 14H10 14C05 14D23 


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© Springer-Verlag Berlin Heidelberg 2016

Authors and Affiliations

  1. 1.Centro de Investigación en MatemáticasGuanajuatoMexico

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