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Here is an example from Ezra Miller's book: Combinatorial Commutative Algebra,p26-27

Let $f,g\in k[x_1,x_2,x_3,x_4]$ be a generic forms of degree $d$ and $e$,the generic initial ideal of $I=\langle f,g\rangle$ for both the lexicographic order and the inverse lexicographic order. When $(d,e)=(2,2)$,the ideals $J=\operatorname{gin}_{\operatorname{lex}}(I)$ are $(x_2^4,x_1x_3^2,x_1x_2,x_1^2)$, the ideal $J=\operatorname{gin}_{\operatorname{revlex}}(I)$ are $(x_2^3,x_1x_2,x_1^2)$.

How to find it?

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How is $\operatorname{gin}_{\operatorname{lex}}$ defined? –  Davide Giraudo Apr 8 '12 at 8:51
Rough speaking,lex is a kind of order <,the initial ideal in$_<$(gI) as a function of g is a constant on a Zariski open subset U in GL$_n$ is called a gin for I and <.I think it is a mainly part of the initial ideal,but I just learn it some days ago,so cannot say anything deeply. –  Strongart Apr 9 '12 at 10:44

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