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

Fix integers $r,s_{1},\dots ,s_{l}$ such that $1\leq l\leq r-1$ and $s_{l}\geq r-l+1$, and let $\mathcal{C}(r;s_{1},\dots ,s_{l})$ 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 $i=2,\dots ,l$, $C$ 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 $(r;s_{1},\dots ,s_{l})$ if $C$ belongs to $\mathcal{C}(r;s_{1},\dots ,s_{l})$. Define $G(r;s_{1},\dots ,s_{l})=\operatorname{max}\left \{p_{a}(C):\,C\in \mathcal{C}(r;s_{1},\dots ,s_{l})\right \},$ where $p_{a}(C)$ denotes the arithmetic genus of $C$. In the present paper, under the hypothesis $s_{1}\gg \dots \gg s_{l}$, we prove that a curve $C$ satisfying the flag condition $(r;s_{1},\dots ,s_{l})$ and of maximal arithmetic genus $p_{a}(C)=G(r;s_{1},\dots ,s_{l})$ must lie on a unique flag such as $C=V_{s_{1}}^{1}\subset V_{s_{2}}^{2}\subset \dots \subset V_{s_{l}}^{l}\subset {\mathbf{P}^{r}}$, where, for any $i=1,\dots ,l$, $V_{s_{i}}^{i}$ 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 $(r-i+1;s_{i},\dots ,s_{l})$ and has maximal arithmetic genus $G(r-i+1;s_{i},\dots ,s_{l})$. This proves the existence of a sort of a hierarchical structure of the family of curves with maximal genus verifying flag conditions.