Normal Bundles of Rational Normal Curves on Hypersurfaces

TL;DR

This paper investigates the normal bundles of rational normal curves on hypersurfaces, providing explicit examples, dimension counts, and bounds for various degrees and splitting types, extending previous work to higher degrees.

Contribution

It extends the study of normal bundles of rational normal curves on hypersurfaces to all degrees, offering explicit examples and dimension calculations for their splitting types.

Findings

Explicit examples of hypersurfaces with given splitting types for all e ≥ 2

Dimension counts for hypersurfaces with specified normal bundle splitting types when d ≥ 3

Lower bounds on maximum rank of quadrics with fixed splitting type for d=2

Abstract

Let $C$ be the rational normal curve of degree $e$ in $\mathbb{P}^n$, and let $X\subset \mathbb{P}^n$ be a degree $d\ge 2$ hypersurface containing $C$. In previous work, I. Coskun and E. Riedl showed that the normal bundle $N_{C/X}$ is balanced for a general $X$. H. Larson studied the case of lines ($e=1$) and computed the dimension of the space of hypersurfaces for which $N_{C/X}$ has a given splitting type. In this paper, we work with any $e\ge 2$. We compute explicit examples of hypersurfaces for all possible splitting types, and for $d\ge 3$, we compute the dimension of the space of hypersurfaces for which $N_{C/X}$ has a given splitting type. For $d=2$, we give a lower bound on the maximum rank of quadrics with fixed splitting type.