A Vanishing Theorem for the Twisted Normal Bundle of Curves in $\mathbb{P}^{n}$, $n\geqslant 8$. Issue 1 (30th March 2020)

• Record Type: Journal Article
• Authors: Ballico, E.
• Is Part Of: Canadian mathematical bulletin =. Volume 63:Issue 1(2020)
• Page Start: 1
• Page End: 5
• Publication Date: 2020-03-30
• Subjects: 14H50; projective curve, -- normal bundle, -- twisted normal bundle; Mathematics -- Periodicals; Mathematics; Periodicals; 510.5
• DOI: 10.4153/S0008439519000146
• Languages: English
• ISSNs: 0008-4395

Abstract

Abstract: 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 $h^{1}(N_{X}(-1))=0$ .