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Extendable codimension two subvarieties in a general hypersurface

G. V. Ravindra, Debaditya Raychaudhury

TL;DR

The paper investigates extendability of codimension $2$ ACM subvarieties inside general hypersurfaces of dimension at least $4$, linking geometric extendability to Hartshorne–Serre constructions and ACM bundle splitting. It combines Hartshorne–Serre correspondence with Green's Koszul exactness and the Beauville–Mérindol criterion to prove that under a concrete numerical condition $\binom{e+5}{4} \le 2d-4$ and suitable generation hypotheses, a codimension-$2$ ACM subvariety $Z\subset X$ is extendable to a codimension-$2 subscheme of $\mathbb{P}^{n+1}$, and that ACM bundles on $X$ with $c_1=\mathcal{O}_X(e)$ often split. The results yield, in particular, nonexistence of globally generated indecomposable ACM bundles of any rank on general hypersurfaces when $e$ is small relative to $d$. The approach unifies Noether–Lefschetz type arguments, Serre correspondence, Green’s Koszul exactness, and splitting criteria to derive extendability and splitting phenomena with clear numerical bounds. These findings provide a broad framework supporting and extending prior work of Voisin, Buchweitz–Greuel–Schreyer, and related results on ACM and Ulrich theories for hypersurfaces.

Abstract

We exhibit a class of extendable codimension $2$ subvarieties in a general hypersurface of dimension at least $4$ in projective space. As a consequence, we prove that a general hypersurface of degree $d$ and dimension at least $4$ does not support globally generated indecomposable ACM bundles of any rank if their first Chern class $e \ll d$.

Extendable codimension two subvarieties in a general hypersurface

TL;DR

The paper investigates extendability of codimension ACM subvarieties inside general hypersurfaces of dimension at least , linking geometric extendability to Hartshorne–Serre constructions and ACM bundle splitting. It combines Hartshorne–Serre correspondence with Green's Koszul exactness and the Beauville–Mérindol criterion to prove that under a concrete numerical condition and suitable generation hypotheses, a codimension- ACM subvariety is extendable to a codimension-\mathbb{P}^{n+1}Xc_1=\mathcal{O}_X(e)ed$. The approach unifies Noether–Lefschetz type arguments, Serre correspondence, Green’s Koszul exactness, and splitting criteria to derive extendability and splitting phenomena with clear numerical bounds. These findings provide a broad framework supporting and extending prior work of Voisin, Buchweitz–Greuel–Schreyer, and related results on ACM and Ulrich theories for hypersurfaces.

Abstract

We exhibit a class of extendable codimension subvarieties in a general hypersurface of dimension at least in projective space. As a consequence, we prove that a general hypersurface of degree and dimension at least does not support globally generated indecomposable ACM bundles of any rank if their first Chern class .

Paper Structure

This paper contains 7 sections, 15 theorems, 79 equations.

Table of Contents

  1. Introduction
  2. Statements of the main results
  3. Preliminaries on Hartshorne-Serre correspondence
  4. Equivalent characterizations of extendability
  5. Surjectivity via Green's theorem
  6. Proof of Theorem \ref{['mainthm2']} via the Beauville-Mérindol criterion
  7. Proofs of Theorem \ref{['mainthm1']} and Theorem \ref{['ggextendable']}

Key Result

Theorem 1

Let $X\subset {\mathbb P}^{n+1}$ be a general hypersurface of dimension $n\geq 4$ and degree $d$. A local complete intersection, ACM codimension $2$ subvariety $Z\subset X$ is extendable if there exists a positive integer $e$ such that The same conclusion holds when $\dim{X}=3$ if condition $(ii)$ above is replaced by the stronger condition that ${I}_{Z/{\mathbb P}^4}(e)$ is globally generated.

Theorems & Definitions (37)

  • Definition 1
  • Remark 1
  • Theorem 1
  • Remark 2
  • Theorem 2
  • Theorem 3
  • Theorem 4: arrondo
  • Remark 3
  • Remark 4
  • Proposition 1
  • ...and 27 more