Hilbert stability of rank-two bundles on curves

Abstract

Let k be an algebraically closed field, and let d and g be two integers with g > 2 and d ^ 1000g(g -1).Let n = d + 2 -2g, and let PTbe a vector space of dimension n.G will denote the grassmannian of all codimension-two subspaces of W, and & will denote the universal rank-two bundle on G.In this paper, a curve will be a connected one-dimensional protective scheme.Let C be a curve on G, i.e., C is a subscheme of G which is a curve, and consider £ = S c = S |C .Let P c (m) = χ((det ί)®") be the Hubert polynomial of C where det E = Λ 2 2s.We let 5 gjrf be the set of all curves C on G with P c (m) -dm + 2 -2g.Thus 5 g ^ is the set of all curves of genus g and degree rfonG.Now W is identified with H°(G 9 S), so given C E S^, there is a natural map φ x :W->H°(C 9 E).We will identify W with H°(C, E) if φj is an isomorphism.Thus we obtain a map ψ2 :A 2 W^H°(C,A 2 E).So for any positive integer m, we obtain a map φ 3 :S m (A 2 w) -*H°{C, (det £) 0m ).We may and do choose m so that <p 3 is onto, so that λ°(C, (det £)® m ) = P c (m) for any C E S gd .Thus we finally obtain a map Λ r(ΛV)-> Λ H°(c,(dQtE)® m ) =k.We say C C G is m-Hilbert stable (resp., m-Hilbert semistable) if φ™ is properly stable (resp., semistable) under the induced action of SL(W) in the