A Vanishing Theorem for the Twisted Normal Bundle of Curves in $\mathbb{P}^{n}$, $n\geqslant 8$
Canadian mathematical bulletin, Tome 63 (2020) no. 1, pp. 1-5
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We prove the existence of a smooth and non-degenerate curve $X\subset \mathbb{P}^{n}$, $n\geqslant 8$, with $\deg (X)=d$, $p_{a}(X)=g$, $h^{1}(N_{X}(-1))=0$, and general moduli for all $(d,g,n)$ such that $d\geqslant (n-3)\lceil g/2\rceil +n+3$. It was proved by C. Walter that, for $n\geqslant 4$, the inequality $2d\geqslant (n-3)g+4$ is a necessary condition for the existence of a curve with $h^{1}(N_{X}(-1))=0$.
Ballico, E. A Vanishing Theorem for the Twisted Normal Bundle of Curves in $\mathbb{P}^{n}$, $n\geqslant 8$. Canadian mathematical bulletin, Tome 63 (2020) no. 1, pp. 1-5. doi: 10.4153/S0008439519000146
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author = {Ballico, E.},
title = {A {Vanishing} {Theorem} for the {Twisted} {Normal} {Bundle} of {Curves} in $\mathbb{P}^{n}$, $n\geqslant 8$},
journal = {Canadian mathematical bulletin},
pages = {1--5},
year = {2020},
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doi = {10.4153/S0008439519000146},
url = {http://geodesic.mathdoc.fr/articles/10.4153/S0008439519000146/}
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