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Moduli of linear slices of high degree hypersurfaces

Anand Patel, Eric Riedl, Dennis Tseng

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.

Moduli of linear slices of high degree hypersurfaces

Abstract

We study the variation of linear sections of hypersurfaces in . 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 hypersurface in vary maximally for . In the process, we generalize the classical Grauert-Mulich theorem about lines in projective space, both to -planes in projective space and to free rational curves on arbitrary varieties.

Paper Structure

This paper contains 11 sections, 28 theorems, 61 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Methods
  3. Acknowledgements
  4. Preliminaries
  5. Notation and Conventions
  6. The map to moduli $\Phi$
  7. Log tangent sheaves
  8. Grauert-Mulich
  9. Plane curves
  10. Hyperplane sections
  11. Bounds and computations for $B_k(n)$

Key Result

Theorem 1.2

If $X\subset \mathbb{P}^2_{\mathbb{C}}$ is an arbitrary plane curve and if $\phi$ fails to have maximal rank, then $X$ has infinitely many projective automorphisms.

Figures (1)

  • Figure 1: Singular curves for which $\phi$ fails to have maximal rank

Theorems & Definitions (52)

  • Theorem 1.2
  • Theorem 1.3
  • Theorem 1.4
  • Lemma 2.1
  • proof
  • Proposition 3.1
  • Definition 3.2
  • Proposition 3.3
  • Lemma 3.4: Descente Lemma from OSS80
  • Lemma 3.5
  • ...and 42 more