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


Author: Vincenzo Di Gennaro
Journal: Proc. Amer. Math. Soc. 136 (2008), 791-799
MSC (2000): Primary 14N15, 14H99; Secondary 14N30, 14M05
DOI: https://doi.org/10.1090/S0002-9939-07-09123-X
Published electronically: November 9, 2007
MathSciNet review: 2361850
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Abstract: 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.


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Additional Information

Vincenzo Di Gennaro
Affiliation: Dipartimento di Matematica, Università di Roma Tor Vergata, Via della Ricerca Scientifica, 00133 Roma, Italia
Email: digennar@axp.mat.uniroma2.it

DOI: https://doi.org/10.1090/S0002-9939-07-09123-X
Keywords: Complex projective curve, Castelnuovo-Halphen theory, arithmetically Cohen-Macaulay curve, arithmetic genus, flag condition, adjunction formula
Received by editor(s): April 21, 2005
Received by editor(s) in revised form: October 15, 2006
Published electronically: November 9, 2007
Communicated by: Michael Stillman
Article copyright: © Copyright 2007 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

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