The general problem of computing invariants of the lexicographic generic initial ideal of an ideal is discussed in the last chapter of [Mark Green Barcelona's notes](http://link.springer.com/chapter/10.1007%2F978-3-0346-0329-4_2#page-1)



There are papers devoted in particular to the study of gin-lex of ideals defining curves  (not only complete intersections) see for example [math/0402418][1] and [arXiv:1108.0045][2].  


  [1]: http://arxiv.org/abs/math/0402418
  [2]: http://arxiv.org/abs/1108.0045