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On Chow stability for algebraic curves

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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 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.

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  1. Centro de Investigación en Matemáticas, Apdo. Postal 402, C.P. 36023, Guanajuato, Gto, Mexico

    L. Brambila-Paz & H. Torres-López

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  1. L. Brambila-Paz
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  2. H. Torres-López
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Correspondence to L. Brambila-Paz.

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Brambila-Paz, L., Torres-López, H. On Chow stability for algebraic curves. manuscripta math. 151, 289–304 (2016). https://doi.org/10.1007/s00229-016-0843-1

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