Regular and rigid curves on some Calabi-Yau and general-type complete intersections

TL;DR

This paper constructs balanced rational and rigid curves of high degrees on certain Calabi-Yau and general-type complete intersections, and applies these results to build rigid bundles on Calabi-Yau threefolds.

Contribution

It introduces methods to construct balanced rational and rigid curves on specific complete intersections, extending the understanding of their geometry and applications.

Findings

Constructed balanced rational curves of all high degrees on certain hypersurfaces and complete intersections.

Built rigid curves of genus g on these varieties for high degrees, especially when n=3 or g=1.

Applied constructions to produce rigid bundles on Calabi-Yau threefolds.

Abstract

Let $X$ be either a general hypersurface of degree $n+1$ in $\mathbb P^n$ or a general $(2,n)$ complete intersection in $\mathbb P^{n+1}, n\geq 4$. We construct balanced rational curves on $X$ of all high enough degrees. If $n=3$ or $g=1$, we construct rigid curves of genus $g$ on $X$ of all high enough degrees. As an application we construct some rigid bundles on Calabi-Yau threefolds. In addition, we construct some low-degree balanced rational curves on hypersurfaces of degree $n + 2$ in $\mathbb P^n$.