An interpolation problem for the normal bundle of curves of genus $g\ge 2$ and high degree in $\mathbb {P}^r$

TL;DR

This paper investigates the strong interpolation property of the normal bundle of high-degree curves of genus at least 2 in projective space, establishing conditions under which this property holds for certain classes of curves.

Contribution

It proves that the normal bundle satisfies strong interpolation for either linearly normal elliptic curves or general high-degree embeddings of curves with genus at least 2.

Findings

Normal bundle satisfies strong interpolation for linearly normal elliptic curves.

Normal bundle satisfies strong interpolation for general embeddings of degree ≥ (5n-8)g + 2n^2 - 5n + 4.

Results extend understanding of normal bundle properties for high-degree curves in projective space.

Abstract

Let $C\subset \mathbb {P}^n$ be a smooth curve and $N_C$ its normal bundle. $N_C$ satisfies strong interpolation if for all integers $s>0$ and $\lambda _i\in \{0,1,\dots ,n-1\}$, $1\le i \le s$, there are distinct points $P_1,\dots ,P_s\in C$ and linear subspaces $U_i\subseteq E|P_i$ such that $\dim (U_i)= \lambda _i$ for all $i$ and the evaluation map $H^0(E)\to \oplus _{i=1}^{s} U_i$ has maximal rank (A. Atanasios). We prove that $C$ satisfies strong interpolation if either $C$ is a linearly normal elliptic curve or $C$ is a general embedding of degree $d\ge (5n-8)g+2n^2-5n+4$ of a smooth curve $X$ of genus $g\ge 2$.