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 ⊂ P n . Namely, if the restriction T P n |C of the tangent bundle of P n to C is stable then C ⊂ 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 P n containing the generic smooth Chow-stable curve of genus g ≥ 4 and degree d > g + n − g n+1 . Moreover, we describe the quotient stack of such curves. Similar results are obtained for the locus of Hilbert stable curves.

On linear stability and syzygy stability

“…We remark that linear stability was introduced by Mumford as it implies Chow stability of the corresponding point in the Hilbert scheme (cf. [2,6]). We will not use Chow stability in this work, however we notice that we have the following: Corollary 3.7 Let C be a curve of genus g ⩾ 11 and gonality ⩾ 5 , and let L be a general element in any component of W 2 g (C) .…”

“…Let C be a smooth projective curve over ℂ , and let L be a globally generated line bundle on C, with deg L = d and dim L = h 0 (C, L) − 1 , let V ⊆ H 0 (C, L) a subspace of dimension r + 1 generating L. Then M V,L ∶= ker(V ⊗ O C → L) is a rank r vector bundle, which appears in different ways and has been given different names in the literature (cf. [4,6,7,9,[11][12][13]).…”

Higher rank Brill-Noether theory and coherent systems: Open questions

“…An interesting concept, which dates back to Mumford, is that of linear stability for a generated rank 1 coherent system (E, V ) ∈ G(1, d, k). Linear stability is linked with the Chow stability of the image of C in P k−1 under the map defined by (E, V ); for some recent work on this link, see [BT16]. Stability (semistability) of D E,V implies linear stability (semistability) of (E, V ), but the converse is not clear.…”

On linear stability and syzygy stability for rank 2 linear series

“…Then M V,L := ker(V ⊗ O C → L) is a rank r vector bundle, it appears in different ways and has been given different names in the literature (cf. [EL89], [But97], [Mis06], [Mis08], [Mis19] [BBN15], [BT16]).…”