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* and *arXiv:1108.0045*.