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

Abstract

Fix integers $r,s_1,...,s_l$ such that $1\leq l\leq r-1$ and $s_l\geq r-l+1$, and let $\Cal C(r;s_1,...,s_l)$ be the set of all integral, projective and nondegenerate curves $C$ of degree $s_1$ in the projective space $\bold P^r$, 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 {\it{flag condition}} $(r;s_1,...,s_l)$ if $C$ belongs to $\Cal C(r;s_1,...,s_l)$. Define $ G(r;s_1,...,s_l)=max\left\{p_a(C): C\in \Cal C(r;s_1,...,s_l)\right \}, $ where $p_a(C)$ denotes the arithmetic genus of $C$. In the present paper, under the hypothesis $s_1>>...>>s_l$, we prove that a curve $C$ satisfying the flag condition $(r;s_1,...,s_l)$ and of maximal arithmetic genus $p_a(C)=G(r;s_1,...,s_l)$ 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, for any $i=1,...,l$, $V_{s_i}^i$ denotes an integral projective subvariety of ${\bold P^r}$ 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.