On Fourier-mukai Transform on the Compact Variety of Ruled Surfaces

Let C be a projective irreducible non-singular curve over an algebraic closed field k of characteristic 0. We consider the Jacobian J(C) of C that is a projective abelian variety parametrizing topological trivial line bundles on C. We consider its Brill-Noether loci that corresponds to the varieties of special divisors. The Torelli theorem allows us to recover the curve from its Jacobian as a polarized abelian variety. We approach the same way the problem for the Quot scheme Qd,r,n(C) of degree d quotients of a trivial vector bundle on C, defining Brill-Noether loci, maps of Abel-Jacobi type. We define a polarisation on the compactification RC,d of the variety of ruled surfaces considered as a Quot scheme and we prove an analogous of the Torelli theorem by applying a Fourier-Mukai transform.