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Interpolation for normal bundles of general curves

About this Title

Atanas Atanasov, Eric Larson and David Yang

Publication: Memoirs of the American Mathematical Society
Publication Year: 2019; Volume 257, Number 1234
ISBNs: 978-1-4704-3489-2 (print); 978-1-4704-4951-3 (online)
DOI: https://doi.org/10.1090/memo/1234
Published electronically: January 9, 2019
MSC: Primary 14H99

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Table of Contents

Chapters

  • 1. Introduction
  • 2. Elementary modifications in arbitrary dimension
  • 3. Elementary modifications for curves
  • 4. Interpolation and short exact sequences
  • 5. Elementary modifications of normal bundles
  • 6. Examples of the bundles $N_{C \to \Lambda }$
  • 7. Interpolation and specialization
  • 8. Reducible curves and their normal bundles
  • 9. A stronger inductive hypothesis
  • 10. Inductive arguments
  • 11. Base cases
  • 12. Summary of Remainder of Proof of Theorem 1.2
  • 13. The three exceptional cases
  • A. Remainder of Proof of Theorem
  • B. Code for Chapter 4

Abstract

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, we 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.

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