Theorem 8.29 in "Combinatorial commutative algebra" by Miller and Sturmfels states the upper-semicontinuity property for Groebner deformations (say, over an algebraically closed field with characteristic zero): roughly speaking, that the Betti numbers of the initial submodule $\mathrm{in}K$ (wrt some term order) of a graded submodule $K$ bound from above those of the graded submodule $K$.
Are there any techniques to examine how tight these bounds are?
I would appreciate also pointers to the literature.
