Communications of the
Korean Mathematical Society
CKMS

ISSN(Print) 1225-1763 ISSN(Online) 2234-3024

Article

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Commun. Korean Math. Soc. 2019; 34(1): 105-111

Online first article June 18, 2018      Printed January 1, 2019

https://doi.org/10.4134/CKMS.c180046

Copyright © The Korean Mathematical Society.

A vanishing theorem for reducible space curves and the construction of smooth space curves in the Range C

Edoardo Ballico

University of Trento

Abstract

Let $Y\subset \cal{P}^3$ be a degree $d$ reduced curve with only planar singularities. We prove that $h^i(\cal{I}_Y(t)) =0$, $i=1,2$, for all $t\ge d-2$. We use this result and linkage to construct some triples $(d,g,s)$, $d>s^2$, with very large $g$ for which there is a smooth and connected curve of degree $d$ and genus $g$, $h^0(\cal{I}_C(s)) =1$ and describe the Hartshorne-Rao module of $C$.

Keywords: space curve, Hilbert function, Halphen's gaps, linkage, Hart\-shorne-Rao module

MSC numbers: 14H50