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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
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.- A. Atanasov, Interpolation and vector bundles on curves, Preprint, math.AG:1404.4892 (2014).
- Maxime Bôcher, The theory of linear dependence, Ann. of Math. (2) 2 (1900/01), no. 1-4, 81–96. MR 1503482, DOI 10.2307/2007186
- Izzet Coskun, Degenerations of surface scrolls and the Gromov-Witten invariants of Grassmannians, J. Algebraic Geom. 15 (2006), no. 2, 223–284. MR 2199064, DOI 10.1090/S1056-3911-06-00426-7
- Lawrence Ein and Robert Lazarsfeld, Stability and restrictions of Picard bundles, with an application to the normal bundles of elliptic curves, Complex projective geometry (Trieste, 1989/Bergen, 1989) London Math. Soc. Lecture Note Ser., vol. 179, Cambridge Univ. Press, Cambridge, 1992, pp. 149–156. MR 1201380, DOI 10.1017/CBO9780511662652.011
- David Eisenbud and Joe Harris, On varieties of minimal degree (a centennial account), Algebraic geometry, Bowdoin, 1985 (Brunswick, Maine, 1985) Proc. Sympos. Pure Math., vol. 46, Amer. Math. Soc., Providence, RI, 1987, pp. 3–13. MR 927946, DOI 10.1090/pspum/046.1/927946
- R. Hartshorne and A. Hirschowitz, Smoothing algebraic space curves, Algebraic geometry, Sitges (Barcelona), 1983, Lecture Notes in Math., vol. 1124, Springer, Berlin, 1985, pp. 98–131. MR 805332, DOI 10.1007/BFb0074998
- Robin Hartshorne, Algebraic geometry, Springer-Verlag, New York-Heidelberg, 1977. Graduate Texts in Mathematics, No. 52. MR 0463157
- Daniel Perrin, Courbes passant par $m$ points généraux de $\textbf {P}^3$, Mém. Soc. Math. France (N.S.) 28-29 (1987), 138 (French, with English summary). MR 925737
- Ziv Ran, Normal bundles of rational curves in projective spaces, Asian J. Math. 11 (2007), no. 4, 567–608. MR 2402939, DOI 10.4310/AJM.2007.v11.n4.a3
- Gianni Sacchiero, Normal bundles of rational curves in projective space, Ann. Univ. Ferrara Sez. VII (N.S.) 26 (1980), 33–40 (1981) (Italian, with English summary). MR 608295
- Jan Stevens, On the number of points determining a canonical curve, Nederl. Akad. Wetensch. Indag. Math. 51 (1989), no. 4, 485–494. MR 1041502