Many combinatorial invariants of a projective variety -- in particular, its (multigraded) Hilbert series, and all the lesser invariants derived from that -- are invariant under flat degeneration.
Two sorts of degenerations are understood best. The first are Gröbner degenerations, which are widely applicable but have terrible geometric properties -- the components of the special fiber are just projective spaces, but they can be trickily nonreduced (and many in number). The second are SAGBI degenerations, which are nice in that the special fiber is a (possibly abnormal) toric variety, but very few examples are known.
I'll talk a bit about the history of this, and present a new sort of degeneration with properties somewhere between these two: it is very widely applicable, and the special fiber is a reduced union of (possibly abnormal) toric varieties -- specifically, torified simplices.