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

Authors:	Vincenzo Di Gennaro

Affiliation:	Dipartimento di Matematica, Università di Roma Tor Vergata, Via della Ricerca Scientifica, 00133 Roma, Italia

Abstract:	Fix integers $r,s_{1},dots ,s_{l}$ such that 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 of degree $s_{1}$ in the projective space $mathbf{P}^{r}$ , such that, for all , does not lie on any integral, projective and nondegenerate variety of dimension and degree $<s_{i}$ . We say that a curve satisfies the flag condition $(r;s_{1},dots ,s_{l})$ if belongs to $mathcal{C}(r;s_{1},dots ,s_{l})$ . Define $G(r;s_{1},dots ,s_{l})=operatorname{max}left {p_{a}(C):,Cin mathcal{C}(r;s_{1},dots ,s_{l})right },$ where $p_{a}(C)$ denotes the arithmetic genus of . In the present paper, under the hypothesis $s_{1}gg dots gg s_{l}$ , we prove that a curve 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 , $V_{s_{i}}^{i}$ denotes an integral projective subvariety of ${mathbf{P}^{r}}$ of degree $s_{i}$ and dimension , 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.

Keywords:	Complex projective curve Castelnuovo-Halphen theory arithmetically Cohen-Macaulay curve arithmetic genus flag condition adjunction formula

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