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.