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Rigid and stably balanced curves on Calabi-Yau and general-type hypersurfaces

Ziv Ran

Abstract

A curve $C$ on a variety $X$ is stably balanced if the slopes of the Harder-Narasimhan filtration of its normal bundle $N$ are contained in an interval of length 1. For each $d\geq n+1$ we construct some regular families of pairs $(C, X)$ of the expected dimension with $X$ a hypersurface of degree $d$ in $\mathbb P^n$ and $C$ a stably balanced rigid curve on $X$, such that the family of hypersurfaces $X$ is smooth codimension $h^1(N)$ in the space of hypersurfaces.

Rigid and stably balanced curves on Calabi-Yau and general-type hypersurfaces

Abstract

A curve on a variety is stably balanced if the slopes of the Harder-Narasimhan filtration of its normal bundle are contained in an interval of length 1. For each we construct some regular families of pairs of the expected dimension with a hypersurface of degree in and a stably balanced rigid curve on , such that the family of hypersurfaces is smooth codimension in the space of hypersurfaces.
Paper Structure (14 sections, 13 theorems, 77 equations)

This paper contains 14 sections, 13 theorems, 77 equations.

Table of Contents

  1. Known results: interpolation, rigidity, stability
  2. New results: Stable balance, rigid regularity
  3. Methods
  4. Fan and quasi-cone degenerations
  5. Stably balanced bundles
  6. Basics
  7. Modifications
  8. Degeneration
  9. Relation to interpolation
  10. Generalities of deformations
  11. Deformations of pairs
  12. Embedded deformations of pairs
  13. Singular case
  14. Results

Key Result

Theorem 1

For each $d\geq n+1\geq 5$, and each $(e, g)$ in a suitable range depending on $n,d$, a general pair $(C, X)$ where $C$ is a (nonspecial) curve of degree $e$ and genus $g$ in $\mathbb P^n$ and $X$ is a hypersurface of degree $d$ containing $C$ has the properties - the pair $(C,X)$ is rigid-regular;

Theorems & Definitions (26)

  • Theorem
  • Lemma 1
  • proof
  • Lemma 2
  • proof
  • Lemma 3
  • Lemma 4
  • proof
  • Lemma 5
  • proof
  • ...and 16 more