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 my thesis I construct a perfect obstruction theory - which naturally provides a virtual fundamental class - for every moduli space of a-stable coherent systems.

A perfect obstruction theory for moduli of coherent systems / Scattareggia, Giorgio. - (2017 Jul 27).

### A perfect obstruction theory for moduli of coherent systems

#### Abstract

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 my thesis I construct a perfect obstruction theory - which naturally provides a virtual fundamental class - for every moduli space of a-stable coherent systems.
##### Scheda breve Scheda completa Scheda completa (DC)
27-lug-2017
Fantechi, Barbara
Perroni, Fabio
Scattareggia, Giorgio
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Utilizza questo identificativo per citare o creare un link a questo documento: `https://hdl.handle.net/20.500.11767/54164`