On the number of components of the moduli schemes of stable torsion-free sheaves on integral curves

Abstract

Here we give an upper bound for the number of irreducible components of the moduli scheme of stable rank r torsion-free sheaves of fixed degree on the integral curve X. This bound depends only on r, Sing(X),pa(X) and the corresponding number for the rank 1 case. Let X be an integral complete curve of arithmetic genus Pa and geometric genus g. For all integers r, d with r > 1 let M(X, r, d) (or just M(r, d)) be the moduli scheme of stable rank r torsion-free sheaves on X with degree d. For general background on stable vector bundles and sheaves on curves and their moduli schemes, see [N] or [S]. The number of irreducible components of M (r, d) does not depend on d, but only on X and r (see Theorem 1.2). Let n(r, X) (or just n(r)) be the number of irreducible components of M(r, d). Here we want to give upper bounds on n(r) depending only on the singularities of X, the integer r and n(l). This is given by Theorem 0.1 (see the discussion in the last part of Remark 1.4). To state Theorem 0.1 (our main result) we need to introduce a few concepts (e.g. measures for the singularities of X) and fix some notation. Let 7r: C -* X be the normalization (hence g = pa(C)). Set 6 := Pa g. Set o:= Ox. For any x C Sing(X), set Ox := Ox,x and let mx be the maximal ideal of Ox; set kx := Ox/mx. Set fx := dimK(Ext1(kx,Ox)) and f := max{fx}xEsirlg(x). For example, we have fx = 1 if and only if Ox is Gorenstein (see e.g. [Co], Lemma 2.1.4). Let ex be the multiplicity of X at x; set := max(ex) and e := Ex ex. Let F be a rank r torsion-free sheaf on X; the set of all formal isomorphism classes of the germs {FX}xEsirlg(x) will be called the formal isomorphism class of F. The set W of all possible isomorphism classes for rank r torsion-free sheaves around Sing(X) has in a natural way a scheme structure; for any reduced subscheme Q of W, let M(r, d, Q) be the reduced subscheme of M(r, d)r.eci parametrizing sheaves with formal isomorphism type in Q. Let n(r, Q) (or n(r, d, Q)) be the number of irreducible components of M(r, d, Q). By Theorem 1.2 this integer does not depend on d. Our main interest is when Q is an irreducible component of Wr.ec. Here is our main result. Received by the editors November 28, 1994. 1991 Mathematics Subject Classification. Primary 14H60, 14D20, 14B99. This research was partially supported by MURST and GNSAGA of CNR (Italy). The author is a member of Europroj (and its group Vector bundles on curves). ?)1997 Arnericari Mathernatical Society