A stratification of the moduli space of vector bundles on curves

Abstract

IntroductionLet E be a vector bundle of rank 2 on a smooth projective curve C of genus g ≥ 2 over an algebraically closed field K of arbitrary characteristic.The invariantwhere the maximum is taken over all line subbundles L of E, is just the minimum of the self intersection numbers of all sections of the ruled surface P(E) → C. Note that E is stable (respectively semistable) if and only if s 1 (E) ≥ 1 (respectively ≥ 0). According to a Theorem of C. Segre s 1 (E) ≤ g. Moreover, the function s 1 is lower semicontinuous. Thus s 1 gives a stratification of the moduli space M(2, d) of stable vector bundles of rank 2 and degree d on C, into locally closed subsets M(2, d, s) according to the value s of s 1 (E). 1 Let M 1 (E) denote the set of line subbundles of E of maximal degree. The set M 1 (E) can be considered as an algebraic scheme in a natural way. Maruyama proved in [10] (see also [8] for different proofs) the following statement: It is shown inIt is the aim of the present paper, to generalize statements (A) and (B) to vector bundles of arbitrary rank r ≥ 2.Let now E be a vector bundle of rank r ≥ 2 over C. For any integer with 1 ≤ k ≤ r − 1 we definewhere the maximum is taken over all subbundles F of rank k of E. There is also an interpretation of the invariant s k (E) in terms of self intersection numbers in the ruled variety P(E) (see [9]). According to a theorem of Hirschowitz (see [4])Moreover, the function s k is lower semicontinuous. Thus s k gives a stratification of the moduli space M(r, d) of stable vector bundles of rank r and degree on d on C into locally closed subsets M(r, d, k, s) according to the value of s and k. There is a component M 0 (r, d, k, s) of M(r, d, k, s) distinguish by the fact that a general E ∈ M 0 (r, d, k, s) admits a stable subbundle F such that E/F is also stable.In this paper we prove (see Theorem 4.2): non-empty, and its componentThe bound g ≥ r+1 2works for all k, 1 ≤ k ≤ r − 1 simultaneously. For some special k the result is better. For example, for k = 1 or r − 1 statement (C) is valid for all g ≥ 2 (see Remark 3.3). 2Let M k (E) denote the set of maximal subbundles of rank k of E and M k (E) denote the closure of the set of stable subbundles F of rank k of E of maximal degree such that E/F is also stable in M k (E). Also M k (E) can be considered as an algebraic scheme in a natural way and M k (E) is a union of components of M k (E).Theorem 4.4 below says:again under the hypothesis g ≥ Acknowledgement Part of the work was done during a visit of both authors to CIMAT, Gto. Mexico. They wish to acknowledge the hospitality of CIMAT. We would like to thank P.E. Newstead for his comments on the first version of this paper and Remark 4.5. The first author also want to thank P. E. Newstead and B. Russo for previous discussions on the subject and the University of Liverpool where those discussion were done.

Scroll codes over curves of higher genus: reducible and superstable vector bundles

“…the subsheaves F such that E/F is locally free. For the geometry of the set of stable bundles with a prescribed s k , see [1]. If K is algebraically closed, char(K ) = 0, and E is a general stable bundle on C with rank r and degree d, then A. Hirschowitz proved that s k (E) = k(r − k)(g − 1) + , where is the unique integer such that + k(r − k)(g − 1) ≡ kd (mod r ) [12,Remark 3.14].…”

“…If K is algebraically closed, char(K ) = 0, and E is a general stable bundle on C with rank r and degree d, then A. Hirschowitz proved that s k (E) = k(r − k)(g − 1) + , where is the unique integer such that + k(r − k)(g − 1) ≡ kd (mod r ) [12,Remark 3.14]. In particular for a general E we have s 1 …”

“…Set m := (C(F q )). Let b 1 , b 2 , y be non-negative integers satisfying m > (−1 + (g − 1)/ p)b 1 + b 2 , b 1 < p, yb 1 …”

Stable vector bundles on curves: numerical invariants of their extremal subbundles

“…They are important invariants of a stable vector bundle and give a stratification of the moduli scheme M(X; r, d) of all rank r stable vector bundles on X with degree d (see [6] and [4]). The affermative solution of the so-called Lange conjecture proved in [10] or [3] or, under certain assumptions on r and g, in [4], shows that all these strata, for fixed t, are non-empty (at least in characteristic 0). However, there should be further constraints on the orderered set of r − 1 integers (s 1 (E), .…”

“…, s r−1 (E)). For a "general" member, E, of each strata of M(X; r, d) (or of each "good component of each strata") there is a unique rank t subbundle F of E with s t (E) = t (deg(E)) − r(deg(F )) (see [4], [11] and [10]). Sometimes, our examples will have a unique such F ; sometimes such F will not be unique and we will describe explicitly all of them (see Remark 2.4).…”