Degrevlex Gröbner bases of generic complete intersections

In this paper, we study the Hilbert–Samuel function of a generic standard graded K-algebra

KX₁,…,X_n]/(g₁,…,g_m)

when refined by an (?)-adic filtration, ? being a linear form. From this we obtain a structure theorem which describes the stairs of a generic complete intersection for the degree-reverse-lexicographic order. We show what this means for generic standard (or Gröbner) bases for this order; in particular, we consider an “orderly filling up” conjecture, and we propose a strategy for the standard basis algorithm which could be useful in generic-like cases.