The period map for cubic threefolds
  Allcock-Carlson-Toledo defined a period map for cubic threefolds which takes values in a ball quotient of dimension 10. A theorem of Voisin implies that this is an open embedding. We determine its image and show that on the algebraic level this amounts to an identification of the algebra of SL(5,C)-invariant polynomials on the third symmetric power of the dual of the tautological representation of SL(5,C) with an explicitly described algebra of meromorphic automorphic forms on the complex 10-ball. 
Eduard Looijenga, Rogier Swierstra