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

Fix integers $r,s_1,\mathrm{.}\mathrm{.}\mathrm{.},s_l$ such that $1\le l\le r-1$ and $s_l\ge r-l+1$, and let $\text{Cal}C(r;s_1,\mathrm{.}\mathrm{.}\mathrm{.},s_l)$ be the set of all integral, projective and nondegenerate curves $C$ of degree $s_1$ in the projective space ${\mathbf{P}}^r$, such that, for all $i=2,\mathrm{.}\mathrm{.}\mathrm{.},l$, $C$ 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 {\it{flag condition}} $(r;s_1,\mathrm{.}\mathrm{.}\mathrm{.},s_l)$ if $C$ belongs to $\text{Cal}C(r;s_1,\mathrm{.}\mathrm{.}\mathrm{.},s_l)$. Define $ G(r;s_1,…,s_l)=max\left{p_a©: C\in \Cal C(r;s_1,…,s_l)\right }, $where$p_a©$ denotes the arithmetic genus of $C$. In the present paper, under the hypothesis $s_1>>\mathrm{.}\mathrm{.}\mathrm{.}>>s_l$, we prove that a curve $C$ satisfying the flag condition $(r;s_1,\mathrm{.}\mathrm{.}\mathrm{.},s_l)$ and of maximal arithmetic genus $p_a(C)=G(r;s_1,\mathrm{.}\mathrm{.}\mathrm{.},s_l)$ must lie on a unique flag such as $C=V_{s_1}^{1}\subset V_{s_2}^{2}\subset … \subset V_{s_l}^{l}\subset {\bold P^r}$,where,forany$i=1,…,l$,$V_{s_i}^i$ denotes an integral projective subvariety of ${\mathbf{P}}^r$ of degree $s_i$ and dimension $i$, such that its general linear curve section satisfies the flag condition $(r-i+1;s_i,\mathrm{.}\mathrm{.}\mathrm{.},s_l)$ and has maximal arithmetic genus $G(r-i+1;s_i,\mathrm{.}\mathrm{.}\mathrm{.},s_l)$. This proves the existence of a sort of a hierarchical structure of the family of curves with maximal genus verifying flag conditions.