Interpolation for Normal Bundles of General Curves

Given $n$ general points $p_1, p_2, /ldots, p_n /in /mathbb P^r$, it is natural to ask when there exists a curve $C /subset /mathbb P^r$, of degree $d$ and genus $g$, passing through $p_1, p_2, /ldots, p_n$. In this paper, the authors give a complete answer to this question for curves $C$ with nonspecial hyperplane section.

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Given $n$ general points $p_1, p_2, /ldots , p_n /in /mathbb P^r$, it is natural to ask when there exists a curve $C /subset /mathbb P^r$, of degree $d$ and genus $g$, passing through $p_1, p_2, /ldots , p_n$. In this paper, the authors give a complete answer to this question for curves $C$ with nonspecial hyperplane section. This result is a consequence of our main theorem, which states that the normal bundle $N_C$ of a general nonspecial curve of degree $d$ and genus $g$ in $/mathbb P^r$ (with $d /geq g + r$) has the property of interpolation (i.e. that for a general effective divisor $D$ of any degree on $C$, either $H^0(N_C(-D)) = 0$ or $H^1(N_C(-D)) = 0$), with exactly three exceptions.