Compactified jacobians
  Let J be the jacobian of a reduced projective curve C with nodes only. 1) We give a simple and natural definition for its many compactifications and show the connection with various other definitions appearing in the literature. 2) Among all compactifications we choose one canonical, and define a theta divisor on it. 3) We give two very explicit and simple descriptions of a stratification of this canonical compactification into homogeneous spaces over J. 
Valery Alexeev