Zero-generic initial ideals

Given a homogeneous ideal I of a polynomial ring A = K[X 1 , . . . , X n ] and a monomial order τ , we construct a new monomial ideal of A associated with I . We call it the zero-generic initial ideal of I with respect to τ and denote it with gin0 (I ). When char K = 0, a zero-generic initial ideal is the usual generic initial ideal. We show that gin0 (I ) is endowed with many interesting properties and, quite surprisingly, it also satisfies Green’s Crystallization Principle, which is known to fail in positive characteristic. Thus, zero-generic initial ideals can be used as formal analogues of generic initial ideals computed in characteristic 0.