On bundles with too many sections on algebraic curves
Peter Newstead

The dimension of the space of linear systems of given degree and dimension on a general curve is given by a formula due to Brill and Noether. One can consider the corresponding construction in higher rank, where linear systems are replaced by coherent systems. Here it is necessary to introduce a concept of stability, dependent on a real parameter, in order to construct moduli spaces. One can show that every component of every such moduli space has dimension at least equal to a given number, which is dependent only on the genus of the curve, the rank and degree of the bundle and the dimension of the space of sections which we consider. We call this number the Brill-Noether number. One question that arises is whether, on a general curve, it is possible to find moduli spaces of coherent systems whose dimension is strictly greater than the Brill-Noether number, and in particular whether the moduli space can be non-empty when the Brill-Noether number is negative. In this talk I will discuss instances in which the answer is yes and raise some open questions.

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