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.
1 Answer
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 Spolynomial, 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 ynickiBirula 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! $\endgroup$ Commented 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. $\endgroup$ Commented Apr 8, 2015 at 10:55