Hierarchical structure of the family of curves with maximal genus verifying flag conditions
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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.References
- G. Castelnuovo, Ricerche di geometria sulle curve algebriche, Zanichelli, Bologna (1937).
- L. Chiantini, C. Ciliberto, and V. Di Gennaro, The genus of projective curves, Duke Math. J. 70 (1993), no. 2, 229–245. MR 1219813, DOI 10.1215/S0012-7094-93-07003-2
- Luca Chiantini, Ciro Ciliberto, and Vincenzo Di Gennaro, On the genus of projective curves verifying certain flag conditions, Boll. Un. Mat. Ital. B (7) 10 (1996), no. 3, 701–732 (English, with Italian summary). MR 1411524
- Ciro Ciliberto, Hilbert functions of finite sets of points and the genus of a curve in a projective space, Space curves (Rocca di Papa, 1985) Lecture Notes in Math., vol. 1266, Springer, Berlin, 1987, pp. 24–73. MR 908707, DOI 10.1007/BFb0078177
- Vincenzo Di Gennaro, A bound on the geometric genus of projective varieties verifying certain flag conditions, Trans. Amer. Math. Soc. 349 (1997), no. 3, 1121–1151. MR 1390976, DOI 10.1090/S0002-9947-97-01785-6
- Joe Harris, Curves in projective space, Séminaire de Mathématiques Supérieures [Seminar on Higher Mathematics], vol. 85, Presses de l’Université de Montréal, Montreal, Que., 1982. With the collaboration of David Eisenbud. MR 685427
- L. Gruson, R. Lazarsfeld, and C. Peskine, On a theorem of Castelnuovo, and the equations defining space curves, Invent. Math. 72 (1983), no. 3, 491–506. MR 704401, DOI 10.1007/BF01398398
- Laurent Gruson and Christian Peskine, Genre des courbes de l’espace projectif, Algebraic geometry (Proc. Sympos., Univ. Tromsø, Tromsø, 1977) Lecture Notes in Math., vol. 687, Springer, Berlin, 1978, pp. 31–59 (French). MR 527229
- G. Halphen, Mémoire sur la classification des courbes gauches algébriques, Oeuvres Complètes, vol. III; also J. École Polytechnique 52 (1882), 1-200.
- M. Noether, Zur Grundlegung der Theorie der algebraischen Raumcurven, Königlichen Akad. der Wissenschaften (1883).
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
- 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
- © Copyright 2007
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication. - 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
- MathSciNet review: 2361850