**Memoirs of the American Mathematical Society**

2019;
105 pp;
Softcover

MSC: Primary 14;

Print ISBN: 978-1-4704-3489-2

Product Code: MEMO/257/1234

List Price: $81.00

AMS Member Price: $48.60

MAA Member Price: $72.90

**Electronic ISBN: 978-1-4704-4951-3
Product Code: MEMO/257/1234.E**

List Price: $81.00

AMS Member Price: $48.60

MAA Member Price: $72.90

# Interpolation for Normal Bundles of General Curves

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*Atanas Atanasov; Eric Larson; David Yang*

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.

#### Table of Contents

# Table of Contents

## Interpolation for Normal Bundles of General Curves

- Cover Cover11
- Title page i2
- Chapter 1. Introduction 18
- Chapter 2. Elementary modifications in arbitrary dimension 512
- Chapter 3. Elementary modifications for curves 1724
- Chapter 4. Interpolation and short exact sequences 2128
- Chapter 5. Elementary modifications of normal bundles 2936
- Chapter 6. Examples of the bundles 𝑁_{𝐶→Λ} 3138
- Chapter 7. Interpolation and specialization 3542
- Chapter 8. Reducible curves and their normal bundles 3744
- Chapter 9. A stronger inductive hypothesis 4552
- Chapter 10. Inductive arguments 4956
- Chapter 11. Base cases 6168
- Chapter 12. Summary of Remainder of Proof of Theorem 1.2 6572
- Chapter 13. The three exceptional cases 6774
- Appendix A. Remainder of Proof of Theorem 1.2 7178
- Appendix B. Code for Chapter 4 93100
- Bibliography 105112
- Back Cover Back Cover1118