Semistable sheaves on projective varieties and their restriction to curves

Abstract

Let X be a nonsingular projective variety of dimension n over an algebraically closed field k. Let H be a very ample line bundle on X. If V is a torsion free coherent sheaf on X we define deg V to be cl(V), c~(H)"~ and/~(V) = deg V/rk V. We call V sernistable (resp. stable) if for all proper subsheaves W of V we have #(W) < #(V) [resp. #(W) < #(V)] (cf. [7, 14]). In this paper we prove that if V is semistable on X then its restriction to a general complete intersection curve of sufficiently high degree is semistable (Theorem 6.1). To give an idea of the proof assume X is a surface and V a vector bundle of rank 2. The restriction of V to a general curve C" of degree m is not semistable if and only if it is not semistable on the generic curve Ym defined over the function field of IPH~ H"). Let/S," be the line bundle on Y,, contradicting the semistability of VI Y,, (cf. Sects. 4.1 and 4.2). First we show that L,, extends uniquely to a line bundle L m on X (Proposition 2.1). If we can get L" as a subbundle of V we are through, for then L" would contradict the semistability of V. So we would like the restriction map H~ Hom(Lm, V))~H~ Hom(L m, V)) to be surjective. Now for fixed L it follows from the lemma of Enriques-Severi (Proposition 3.2; [6, Corollary 7.8]) that H~ Horn(L, V))~H~ Horn(L, V)) is surjective for large m. Therefore it is enough if the L" remain the same line bundle L for infinitely many m. To prove that L,, = L we construct a degenerating family of curves D f ~S, X x S 3 D p ~X, such that the generic fibre is a curve Ct"+ 1) of degree 2 "+ 1 and the special fibre is a reduced curve with two nonsingular components CI") of degree 2" (cf. Sect. 5). Let (m) denote 2". Extending the subbundle Lt,,+~)[Ct"+~) to a subsheaf of p*(V) on D and restricting the extension to CI") gives a lower bound for the maximal degree of a line subbundle of V[CIm ) in terms of that for V[Ctm+I ) (Proposition 4.3). This implies that degL,, is bounded (Lemma 6.5.1) so that for an infinite subsequence of m, degL,, is constant. If degLtm + r)= degLt,,) by refining the above argument with the degenerating family one can prove that Lt"+,)[ CI")