Density and completeness of subvarieties of moduli spaces of curves or Abelian varieties

Let Mg be the moduli space of smooth curves of genus g ≥ 2 and let Ag be the moduli space of principally polarized abelian varieties (ppav) of dimension g over C. A (Deligne-Mumford) stable curve of genus g is a reduced, connected and complete curve of arithmetic genus g with only nodes as singularities and with finite automorphism group. We say that a stable curve is of compact type if its generalized jacobian is an abelian variety. We denote by M̃g the moduli space of stable curves of compact type and genus g over C. By “density” we always mean “analytic density” unless we specify otherwise. Given a subvariety V of Mg or M̃g and an integer q between 1 and g/2, let Eq (V ) be the subset of V parametrizing curves whose jacobian contains an abelian variety of dimension q . We define Eq (V ) for V a subvariety of Ag in a similar fashion. It is well-known that Eq (Ag) is dense in Ag for all q . Colombo and Pirola pose the following question in [3]