Moduli of linear slices of high degree hypersurfaces

TL;DR

This paper investigates how linear sections of high-degree hypersurfaces vary in moduli, classifies special plane curves with non-maximal variation, and generalizes classical theorems to higher-dimensional cases.

Contribution

It provides a complete classification of certain singular plane curves with non-maximal moduli variation and proves maximal variation of hyperplane sections for smooth hypersurfaces of degree at least n+3.

Findings

Classified all singular plane curves with non-maximal moduli variation.

Proved maximal variation of hyperplane sections for smooth degree d hypersurfaces with d ≥ n+3.

Generalized Grauert-Mulich theorem to higher-dimensional projective spaces and rational curves.

Abstract

We study the variation of linear sections of hypersurfaces in $\mathbb{P}^n$. We completely classify all plane curves, necessarily singular, whose line sections do not vary maximally in moduli. In higher dimensions, we prove that the family of hyperplane sections of any smooth degree $d$ hypersurface in $\mathbb{P}^n$ vary maximally for $d \geq n+3$. In the process, we generalize the classical Grauert-Mulich theorem about lines in projective space, both to $k$-planes in projective space and to free rational curves on arbitrary varieties.