Hierarchical structure of the family of curves with maximal genus verifying flag conditions

Fix integers r, s(1),..., s(l) such that 1 <= l <= r-1 and s(l) >= r-l+1, and let C(r; s(1),..., s(l)) be the set of all integral, projective and nondegenerate curves C of degree s(1) in the projective space P-r, such that, for all i = 2,..., l, C does not lie on any integral, projective and nondegenerate variety of dimension i and degree < s(i). We say that a curve C satisfies the flag condition (r; s(1),..., s(l)) if C belongs to C(r; s(1),..., s(l)). Define G(r; s(1),..., s(l)) = max {pa(C) : C is an element of C( r; s(1),..., s(l))}, where p(a)(C) denotes the arithmetic genus of C. In the present paper, under the hypothesis s(1) >> center dot center dot center dot >> s(l), we prove that a curve C satisfying the flag condition (r; s(1),..., s(l)) and of maximal arithmetic genus pa(C) = G(r; s(1),..., s(l)) must lie on a unique flag such as C = V-s1(1) subset of V-s2(2) subset of center dot center dot center dot subset of V-sl(l) subset of P-r, where, for any i = 1,..., l, V-si(i) denotes an integral projective subvariety of P-r of degree s(i) and dimension i, such that its general linear curve section satisfies the flag condition (r-i+1; s(i),..., s(l)) and has maximal arithmetic genus G(r-i+1; s(i),..., s(l)). This proves the existence of a sort of a hierarchical structure of the family of curves with maximal genus verifying flag conditions.