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The Noether--Lefschetz theorem in arbitrary characteristic

Lena Ji

Abstract

We show that if $X\subset\mathbb P^N_k$ is a normal variety of dimension $\geq 3$ and $H\subset\mathbb P^N_k$ a very general hypersurface of degree $d=4$ or $\geq 6$, then the restriction map $\mathrm{Cl}(X)\to\mathrm{Cl}(X\cap H)$ is an isomorphism up to torsion. If $\dim X\geq 4$, the result holds for $d\geq 2$. The proof uses the relative Jacobian of a curve fibration, together with a specialization argument, and the result holds over fields of arbitrary characteristic.

The Noether--Lefschetz theorem in arbitrary characteristic

Abstract

We show that if is a normal variety of dimension and a very general hypersurface of degree or , then the restriction map is an isomorphism up to torsion. If , the result holds for . The proof uses the relative Jacobian of a curve fibration, together with a specialization argument, and the result holds over fields of arbitrary characteristic.

Paper Structure

This paper contains 16 sections, 35 theorems, 23 equations.

Table of Contents

  1. Introduction
  2. Comparison with previously known results
  3. Outline
  4. Preliminaries
  5. Class groups of normal varieties
  6. Very general
  7. Thin sets and Néron's theorem
  8. Bertini theorems
  9. Injectivity
  10. Sections of the relative Jacobian
  11. The connected component $\mathop{\mathrm{Cl}}\nolimits^0$
  12. Injectivity for $\mathop{\mathrm{Cl}}\nolimits$
  13. Surjectivity
  14. Quartic surfaces in $\mathbb P^3$
  15. Surjectivity over fields of infinite transcendence degree
  16. ...and 1 more sections

Key Result

Theorem 1.1

Let $k$ be an algebraically closed field, $X$ a normal projective variety of dimension $\geq 3$ over $k$, $\mathcal{L}_0$ a very ample line bundle on $X$, and $d\in\mathbb Z$. For $Y\in|\mathcal{L}_0^{\otimes d}|$ the restriction map \xymatrix{ \mathop{\mathrm{Cl}}\nolimits(X) \ar[r] & \mathop{\math

Theorems & Definitions (61)

  • Theorem 1.1: Theorems \ref{['prop:injectivity']} and \ref{['prop:surjectivity']}
  • Definition 2.1: kol_mumford
  • Theorem 2.2: [neron52; lang-neron; kol_mumford, Theorem 17; kol_determines, Section 14]
  • Lemma 2.3: kol_mumford
  • Lemma 2.4
  • proof
  • Definition 2.5: weil-foundations
  • Definition 2.6: [serre92, Chapter 3; fm08, Chapter 12]
  • Theorem 2.7: neron52
  • Theorem 2.8: fov99
  • ...and 51 more