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

TL;DR

This paper constructs families of stably balanced, rigid curves on hypersurfaces in projective space, demonstrating their existence on Calabi-Yau and general-type hypersurfaces with controlled deformation properties.

Contribution

It introduces explicit constructions of stably balanced rigid curves on hypersurfaces of degree d in projective space, expanding understanding of curve behavior on Calabi-Yau and general-type varieties.

Findings

Families of such curves exist for degrees d ≥ n+1.

The constructed families have the expected dimension.

The hypersurfaces form a smooth codimension h^1(N) subset in the space of all hypersurfaces.

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.