Abstract
The definition of a generic initial ideal includes the assumption $x_1>x_2> \cdots >x_n$. A natural question is how generic initial ideals change when we permute the variables. In the article [1, §2], it is shown that the generic initial ideals are permuted in the same way when the variables in the monomial order are permuted. We give a different proof of this theorem. Along the way, we study the Zariski open sets which play an essential role in the definition of a generic initial ideal and also prove a result on how the Zariski open set changes after a permutation of the variables.
Highlights
Let S = F [x1, . . . , xn] be a polynomial ring in n variables where F is an infinite field
Some of the properties of generic initial ideals were used by Hartshorne to prove the connectedness of Hilbert schemes
We may consider the fact that generic initial ideals were exploited to bound the invariants of projective varieties
Summary
Let < be a monomial order on S satisfying x1 > x2 > · · · > xn and in
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