| We study the GIT quotients of the Hilbert and Chow schemes of curves of genus g and degree d in the projective space of dimension d-g, as d decreases with respect to g. We show that the previous results of Lucia Caporaso hold true up to d not higher than 4(2g-2) and we observe that this is sharp. In the range for d higher than 2(2g-2) but smaller than 7/2(2g-2), we get a complete new description of the GIT quotient and observe that the above range is also sharp. As a corollary of our results, we get a new compactification of the universal Jacobian over the Schubert moduli space of pseudo-stable curves. This is a joint work with Gilberto Bini and Margarida Melo. |