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On the Abel-Jacobi map for divisors of higher rank on a curve. (English) Zbl 0840.14003

The aim of this paper is to present an algebro-geometric approach to the study of the geometry of the moduli space of stable bundles on a smooth projective curve defined over an algebraic closed field \(k\), of arbitrary characteristic. One of the basic ideas is to consider a notion of divisor of higher rank and a suitable Abel-Jacobi map generalizing the classical notions in rank one. Let \({\mathcal O}_C\) be the structure sheaf of the curve \(C\) and let \(K\) be its field of rational functions, considered as a constant \({\mathcal O}_C\)-module.
We define a divisor of rank \(r\) and degree \(n\), an \((r,n)\)-divisor for short, to be any coherent sub-\({\mathcal O}_C\)-module of \(K^r= K^{\oplus r}\) having rank \(r\) and degree \(n\). Since \(C\) is smooth, these submodules are locally free and coincide with the matrix divisors defined by A. Weil [J. Math. Pures Appl., IX. Sér. 17, 47-87 (1938; Zbl 0018.06302)]. Denote by \(\text{Div}^{r,n}_{C/k}\) the set of all \((r,n)\)-divisors. This set can be identified with the set of rational points of an algebraic ind-variety \({\mathcal D} iv^{r,n}_{C/k}\) that may be described as follows. For any effective ordinary divisor \(D\) set \(\text{Div}^{r,n}_{C/k} (D)= \{E\in \text{Div}^{r,n}_{C/k}\mid E\subseteq {\mathcal O}_C (D)^r\}\) where \({\mathcal O}_C (D)^r\) is considered as a sub \({\mathcal O}_C\)-module of \(K^r\). The elements of the set \(\text{Div}^{r,n}_{C/k} (D)\) can be identified with the rational points of the scheme \(\text{Quot}^m_{{\mathcal O}_C (D)^r/X/k}\), \(m=r\cdot \deg D-n\), parametrizing torsion quotients of \({\mathcal O}_C (D)^r\) having degree \(m\). It is natural to stratify the ind-variety \({\mathcal D} iv^{r,n}_{C/k}\) according to Harder-Narasimhan type \({\mathcal D} iv^{r,n}_{C/k}= ({\mathcal D} iv^{r,n}_{C/k} )^{ss} \cup \bigcup_{P\neq ss} {\mathcal S}_P\) where \(({\mathcal D} iv^{r,n}_{C/k} )^{ss}\) is the open ind-subvariety of semistable divisors. The cohomology of each stratum stabilizes and this stratification is perfect. (Here cohomology means \(\ell\)-adic cohomology for a suitable prime \(\ell\).) In particular, there is an identity of Poincaré series \[ P({\mathcal D} iv^{r,n}_{C/k}; t)= P\bigl( ({\mathcal D} iv^{r,n}_{C/k} )^{ss}; t\bigr)+ \sum_{P\neq ss} P({\mathcal S}_P, t)\cdot t^{2d_r}, \] where \(d_P\) is the codimension of \({\mathcal S}_P\). Let \(r\) and \(n\) be coprime. Then the notations of stable and semistable bundle over \(C\) coincide, and the moduli space \(N(r,n)\) of stable vector bundles having rank \(r\) and degree \(n\) is in this case a smooth projective algebraic variety. It is natural to define, by analogy with the classical case, Abel-Jacobi maps \(\vartheta: ({\mathcal D} iv^{r,n}_{C/k} )^{ss}\to N(r,n)\), by assigning to a divisor \(E\) its isomorphism class as a vector bundle. In order to find the Betti numbers of \(N(r,n)\) it suffices to know those of \(({\mathcal D} iv^{r,n}_{C/k} )^{ss}\). This computation reduces, for arbitrary \(r\) and \(n\), to the calculation of those of \({\mathcal D} iv^{r,n}_{C/k}\).
The varieties \(\text{Div}^{r,n}_{C/k} (D)\) are analogous to Grassmannians and share with them the property of having a decomposition into Schubert “strata”. One obtains \[ P({\mathcal D} iv^{r,n}_{C/k}; t)= {{\prod^r_{j=1} (1+t^{2j-1} )^{2g}} \over {(1-t^{2r}) \prod^{r-1}_{j=1} (1- t^{2j})^2}}. \]

MSC:

14C20 Divisors, linear systems, invertible sheaves
14H60 Vector bundles on curves and their moduli
13D40 Hilbert-Samuel and Hilbert-Kunz functions; Poincaré series

Citations:

Zbl 0018.06302
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References:

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