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Vector bundles and Brill-Noether theory. (English) Zbl 0866.14024

Clemens, Herbert (ed.) et al., Current topics in complex algebraic geometry. New York, NY: Cambridge University Press. Math. Sci. Res. Inst. Publ. 28, 145-158 (1995).
The moduli spaces of (semi-)stable rank-\(n\) vector bundles on a smooth complex algebraic curve \(X\), with fixed determinant line bundle \((L\in\text{Pic}(X)\), are highly interesting geometric objects in several regards. Apart from their appearance in conformal quantum field theory, they often represent types of algebraic varieties that are difficult to construct by other means. On the other hand, the existence of special points in these moduli spaces (i.e., the existence of special line bundles on \(X)\) provides interesting embeddings of the base curve \(X\) into Grassmannian subvarieties of projective spaces. – The present article surveys some recent developments in this direction, and sketches a few new results obtained by the author himself. The main objects of interest here are rank-two vector bundles with canonical determinant on a curve \(X\), which possess as many global sections as possible.
The paper consists of four sections. In the first two sections, the author gives quick but lucid reviews of the classical results on Jacobians of curves, Brill-Noether theory of special divisors, and the geometry of the Petri map. This serves, in a very enlightening manner, as a motivation for the more recent material discussed in the following two sections. Semistable and stable vector bundles on curves as well as their moduli spaces are discussed in section 3, in particular under the aspect of what these moduli spaces look like in the special case of rank-two bundles with determinant of odd degree. Special emphasis is put on illustrating the “classical” results of S. Ramanan (1973), I. Desale and S. Ramanan (1976), P. E. Newstead (1968), and D. Mumford and P. E. Newstead (1968), by which those moduli spaces are identified as Fano varieties, complete intersections, or Jacobians, respectively.
The concluding section 4 turns to the more recent progress made in this context. The author describes a Brill-Noether theory for vector bundles of rank two and canonical determinant, which differs from the recent (so-called) “higher Brill-Noether theory” developed by other authors and well-known to experts. By these means, the author proves that the Brill-Noether loci defined by bundles with “many” global sections satisfy a certain dimension inequality. Moreover, using the Petri map for those vector bundles, the Zariski cotangent spaces at points of these Brill-Noether loci are computed, and smoothness conditions are derived.
This discussion is enhanced by the statement of five open problems in this context. At the end, the author gives some sample results related to the study of the Grassmann map associated with the linear system of bundles in his Brill-Noether loci. These examples (theorems) are closely related to the author’s classification of Fano threefolds via vector bundles [cf. S. Mukai in: Complex Projective Geometry, Sel. Pap. Conf. Proj. Var., Trieste 1989, and Vector Bundles and Spec. Proj. Embeddings, Bergen 1989, Lond. Math. Soc. Lect. Note Ser. 179, 255-263 (1992; Zbl 0774.14037)] and have been published separately [cf. S. Mukai in: Algebraic Geometry and Related Topics, Proc. internat. Sympos. Incheon, 1992, Conf. Proc. Lect. Notes Algebr. Geom. 1, 19-40 (1993; Zbl 0846.14030)].
Altogether, this brief survey is masterly written and valuable for both experts and non-experts.
For the entire collection see [Zbl 0836.00011].

MSC:

14H60 Vector bundles on curves and their moduli
14M12 Determinantal varieties
14J45 Fano varieties
14D20 Algebraic moduli problems, moduli of vector bundles
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