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Full Description
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.
Contents
Introduction
Elementary modifications in arbitrary dimension
Elementary modifications for curves
Interpolation and short exact sequences
Elementary modifications of normal bundles
Examples of the bundles $N_{C \to \Lambda }$
Interpolation and specialization
Reducible curves and their normal bundles
A stronger inductive hypothesis
Inductive arguments
Base cases
Summary of Remainder of Proof of Theorem 1.2
The three exceptional cases
Appendix A. Remainder of Proof of Theorem 1.2
Appendix B. Code for Section 4
Bibliography.
