Compactification of the moduli space of hyperplane arrangements
  Consider the moduli space M^0 of arrangements of n hyperplanes in general position in projective (r-1)-space. When r=2 the space has a compactification given by the moduli space of stable curves of genus 0 with n marked points. In higher dimensions, the analogue of the moduli space of stable curves is the moduli space of stable pairs: pairs (S,B) consisting of a variety S (possibly reducible) and a divisor B=B_1+..+B_n, satisfying various additional assumptions. We identify the closure of M^0 in the moduli space of stable pairs as Kapranov's Chow quotient compactification of M^0, and give an explicit description of the pairs at the boundary. We also construct additional irreducible components of the moduli space of stable pairs. 
Paul Hacking, Sean Keel, and Jenia Tevelev