Hierarchical structure of the family of curves with maximal genus verifying flag conditions
Abstract
Description
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.
10 pages, revised version, to appear in Proc. Amer. Math. Soc
10 pages, revised version, to appear in Proc. Amer. Math. Soc