Assume a fintie set of monomials is given. Is there a way to find the family of ideals whose initial ideal (say w.r.t revlex order) is generated by that finite set? I'll appreciate any partial answer, reference or suggestions.
It's kinda gross, but it can be done.
To each monomial, add a generic linear combination of all smaller monomials (w.r.t. your term order).
Now insist that what you have is a Gr\"obner basis. How do you do this? Apply the reduction algorithm to each S-polynomial, until you get stuck. Then assert that the result is zero. This puts a multitude of algebraic conditions on the coefficients in your generic linear combinations.
I hope it's obvious that the resulting set thus constructed, a Bia\l ynicki-Birula stratum on the Hilbert scheme, should be termed the "Gr\"obner basin".
$\begingroup$ Thank you for your answer. Theoretically it is true But the problem ist I don't know how to be sure that the combination is a "generic" one in a concrete given situation. However I know that this problem should not be very easily accessible too, since the consequences are very surprising! $\endgroup$ Apr 8, 2015 at 7:12
1$\begingroup$ Sorry, all I mean is that you should put in a coefficient for every possible term, then use the above to constrain the coefficients. This answer makes no assumption of genericity; it really does find every ideal with this initial ideal. $\endgroup$ Apr 8, 2015 at 10:55