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Softcover ISBN:  9781470434892 
Product Code:  MEMO/257/1234 
List Price:  $81.00 
MAA Member Price:  $72.90 
AMS Member Price:  $48.60 
eBook ISBN:  9781470449513 
Product Code:  MEMO/257/1234.E 
List Price:  $81.00 
MAA Member Price:  $72.90 
AMS Member Price:  $48.60 
Softcover ISBN:  9781470434892 
eBook ISBN:  9781470449513 
Product Code:  MEMO/257/1234.B 
List Price:  $162.00 $121.50 
MAA Member Price:  $145.80 $109.35 
AMS Member Price:  $97.20 $72.90 

Book DetailsMemoirs of the American Mathematical SocietyVolume: 257; 2019; 105 ppMSC: Primary 14
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

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


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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.

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