Let $C\subset \bold P^r$ be an integral projective curve. One defines the speciality index $e(C)$ of $C$ as the maximal integer $t$ such that $h^0(C,\omega_C(-t))>0$, where $\omega_C$ denotes the dualizing sheaf of $C$. Extending a classical result of Halphen concerning the speciality of a space curve, in the present paper we prove that if $C\subset \bold P^5$ is an integral degree $d$ curve not contained in any surface of degree $< s$, in any threefold of degree $<t$, and in any fourfold of degree $<u$, and if $d>>s>>t>>u\geq 1$, then $e(C)\leq {\frac{d}{s}}+{\frac{s}{t}}+{\frac{t}{u}}+u-6.$ Moreover equality holds if and only if $C$ is a complete intersection of hypersurfaces of degrees $u$, ${\frac{t}{u}}$, ${\frac{s}{t}}$ and ${\frac{d}{s}}$. We give also some partial results in the general case $C\subset \bold P^r$, $r\geq 3$.

DI GENNARO, V., Franco, D. (2007). A speciality theorem for curves in $bold P^5$. GEOMETRIAE DEDICATA, 129, 88-99.

### A speciality theorem for curves in $bold P^5$.

#### Abstract

Let $C\subset \bold P^r$ be an integral projective curve. One defines the speciality index $e(C)$ of $C$ as the maximal integer $t$ such that $h^0(C,\omega_C(-t))>0$, where $\omega_C$ denotes the dualizing sheaf of $C$. Extending a classical result of Halphen concerning the speciality of a space curve, in the present paper we prove that if $C\subset \bold P^5$ is an integral degree $d$ curve not contained in any surface of degree $< s$, in any threefold of degree $>s>>t>>u\geq 1$, then $e(C)\leq {\frac{d}{s}}+{\frac{s}{t}}+{\frac{t}{u}}+u-6.$ Moreover equality holds if and only if $C$ is a complete intersection of hypersurfaces of degrees $u$, ${\frac{t}{u}}$, ${\frac{s}{t}}$ and ${\frac{d}{s}}$. We give also some partial results in the general case $C\subset \bold P^r$, $r\geq 3$.
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Pubblicato
Rilevanza internazionale
Articolo
Sì, ma tipo non specificato
Settore MAT/03 - Geometria
English
Con Impact Factor ISI
Complex projective curve, speciality index, arithmetic genus, adjunction formula, complete intersection, linkage, Castelnuovo - Halphen Theory, flag conditions.
DI GENNARO, V., Franco, D. (2007). A speciality theorem for curves in $bold P^5$. GEOMETRIAE DEDICATA, 129, 88-99.
DI GENNARO, V; Franco, D
Articolo su rivista
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Utilizza questo identificativo per citare o creare un link a questo documento: https://hdl.handle.net/2108/34846
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