Geometric Invariant Theory (GIT) was introduced by Mumford and co-authors in order to deal with the problem of constructing quotients of algebraic varieties by the action of an algebraic group. It is particularly useful in the construction of moduli spaces within the category of algebraic varieties, which has been the main motivation for the development of the theory. In this talk I will report on a joint work with G. Bini and F. Viviani in which we study the GIT problem for the Hilbert and Chow scheme of curves of genus g embedded via a complete linear system of low degree. We give a characterization of GIT-semistable, polistable and stable points for both the Hilbert and Chow scheme for degree 2(2g-2)4(2g-2). Our description extends the work of L. Caporaso in ''A compactification of the universal Picard variety over the moduli space of stable curves'', which deals with the case d>10(2g-2) for the Hilbert scheme. As a corollary of our analysis, we obtain a new modular compactification of the universal Picard variety over the moduli space of pseudo-stable curves. This construction should also play an important role in the study of the birational geometry of the universal Picard variety.
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