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Subvarieties of low degree on general hypersurfaces

Nathan Chen, David Yang

TL;DR

This work analyzes subvarieties of small degree on very general hypersurfaces of high degree in projective space. It develops incidence-correspondence techniques and Castelnuovo-type genus bounds to show that low-degree curves on a general $X$ of degree $d \ge 2n$ must arise as plane sections when $\delta \le d+2$, and further extends to higher degrees via an inductive framework: for each fixed $s$, large $d$ forces any integral curve of degree at most $sd$ to be a complete intersection with a ambient subvariety of degree at most $s$; a corresponding higher-dimensional analogue (Theorem B) yields a unique ambient subvariety describing $Y = X \cap V$. These results refine understanding of how a very general hypersurface inherits subvarieties from the ambient space, and connect to Noether–Lefschetz-type phenomena and genus bounds for subvarieties.

Abstract

The purpose of this note is to show that the subvarieties of small degree inside a general hypersurface of large degree come from intersecting with linear spaces or other varieties.

Subvarieties of low degree on general hypersurfaces

TL;DR

This work analyzes subvarieties of small degree on very general hypersurfaces of high degree in projective space. It develops incidence-correspondence techniques and Castelnuovo-type genus bounds to show that low-degree curves on a general of degree must arise as plane sections when , and further extends to higher degrees via an inductive framework: for each fixed , large forces any integral curve of degree at most to be a complete intersection with a ambient subvariety of degree at most ; a corresponding higher-dimensional analogue (Theorem B) yields a unique ambient subvariety describing . These results refine understanding of how a very general hypersurface inherits subvarieties from the ambient space, and connect to Noether–Lefschetz-type phenomena and genus bounds for subvarieties.

Abstract

The purpose of this note is to show that the subvarieties of small degree inside a general hypersurface of large degree come from intersecting with linear spaces or other varieties.

Paper Structure

This paper contains 4 sections, 10 theorems, 55 equations.

Table of Contents

  1. Genus bounds for curves on very general hypersurfaces
  2. Incidence correspondences
  3. Proof of Theorem \ref{['thm:planeCurvesMinimal']}
  4. Subvarieties of higher degree

Key Result

Theorem 1

Let $X \subset \mathbb{P}^{n+1}_{\mathbb{C}}$ be a general hypersurface of degree $d \geq 2n$ and let $Y \subset X$ be a positive-dimensional subvariety such that $\deg Y \leq d+2$. Then $\deg Y = d$ and $Y = X \cap \Lambda$ is the intersection of $X$ with a linear subspace $\Lambda \cong \mathbb{P}

Theorems & Definitions (21)

  • Theorem 1
  • Theorem 2
  • Theorem 1.1
  • proof : Proof of Theorem \ref{['thm:main']}
  • Proposition 1.2
  • proof
  • Proposition 1.3
  • proof
  • Proposition 1.4: Castelnuovo
  • Lemma 2.1
  • ...and 11 more