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A perfect obstruction theory for coherent systems

Giorgio Scattareggia
Friday, November 25, 2016 - 11:00

According to the definition introduced by Newstead and Le Potier, a coherent system on a curve is an algebraic vector bundle together with a linear subspace of prescribed dimension of its space of sections. Therefore, coherent systems are a natural generalization of the classical notion of linear series. It is well known that the variety which parametrizes linear series on a fixed genus g curve has expected dimension given by the Brill-Noether number g-(r+1)(g-d+r). An analogous result holds for moduli spaces of stable coherent systems: if a is a real number (which is needed in order to define the notion stability for coherent systems), the moduli space of a-stable coherent systems of rank n, degree d and dimension k has expected dimention given by n^2(g-1) +1 -k(k-d+m(g-1)). Then, what is expected is the existence of a virtual fundamental class on these moduli spaces which justifies their expected dimensions.
In this talk I will show the construction of a perfect obstruction theory - which naturally provides a virtual fundamental class - for a generalization of this moduli problem. Indeed, I will consider couples of algebraic vector bundles on a curve, with a fixed morphism between the two bundles, and I will study the moduli stack/space attached to these objects. The existence of a perfect obstruction theory for coherent systems will follow from this construction.

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