Our goal in this tutorial is to give a quick overview of generic initial ideals. A more comprehensive treatment is in [Gr96]. We first lay out the basic facts and notations, and then deal with a few of the more interesting points in a question and answer format.

We would like to thank Bruno Buchberger for inviting us to contribute this paper, and also for having, through his fundamental contributions, made the work discussed in this tutorial possible.

For this tutorial we let S = C[x1,…, xn]. Some of what we do also works in characteristic p > 0, but the combinatorial properties of Borel fixed monomial ideals are more complicated. Later we give an indication of what is true in this case.

Let I be a homogeneous ideal in the polynomial ring S, and choose a monomial order. Throughout this tutorial, the only monomial orders that we consider satisfy x1 > x2 > … > xn. Any such order will do, but the most interesting from our present point of view are the lexicographic and reverse lexicographic orders. Given this monomial order, we may compute a Gröbner basis {g1,…, gr} of the ideal I, using Buchberger's algorithm. The initial ideal in (I) is the monomial ideal generated by the lead terms of g1,…,gr. This monomial ideal has the same Hilbert function as I.