he speciality theorem of Gruson and Peskine says that if $C \subseteq \bbfP^3$ is a reduced, irreducible curve of degree $d$, not contained in any surface of degree $< t$, and $n$ is an integer such that $H^1 (\bbfP^3, {\cal O}_C (n)) \ne 0$, then $n \le t + (d/t) - 4$, and the equality is attained only if $C$ is a complete intersection (of surfaces of degree $t$ and $d/t)$. In this note, the same theorem is proved with a new, easy proof (based on the use of rank two reflexive sheaves) and which holds also for reducible or not reduced curves.
Some remarks about the speciality theorem of Gruson and Peskine
ROGGERO, Margherita
1985-01-01
Abstract
he speciality theorem of Gruson and Peskine says that if $C \subseteq \bbfP^3$ is a reduced, irreducible curve of degree $d$, not contained in any surface of degree $< t$, and $n$ is an integer such that $H^1 (\bbfP^3, {\cal O}_C (n)) \ne 0$, then $n \le t + (d/t) - 4$, and the equality is attained only if $C$ is a complete intersection (of surfaces of degree $t$ and $d/t)$. In this note, the same theorem is proved with a new, easy proof (based on the use of rank two reflexive sheaves) and which holds also for reducible or not reduced curves.
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