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The relative Fujita-Zariski theorem

Laurent Moret-Bailly

TL;DR

The paper proves a relative version of the Fujita–Zariski theorem: if $\mathscr{L}$ is an invertible sheaf on a proper $X\to \mathrm{Spec}(R)$ whose restriction to the base locus $B$ is ample, then $\mathscr{L}^{\otimes t}$ is globally generated for $t$ sufficiently large. It develops a relative framework using graded cohomology modules $H^q_{*}(\mathscr{F})$ and $\Lambda$-modules (graded $\mathrm{Sym}_R(V)$-submodules) to analyze generation properties, and proves a key finiteness result via a dévissage argument and a fat-subset general-position technique. The proof proceeds by noetherian induction on $X$, selecting general elements of $V$ to control base loci and using exact sequences to lift generation from subschemes to $X$. Together, these methods establish a robust relative base-point freeness criterion and extend the classical result beyond the base-field setting, with potential applicability to algebraic spaces.

Abstract

We prove, with no claim to originality, a relative version of the Fujita-Zariski theorem. When the base is a field, this result is due to Fujita (1983) and states that if an invertible sheaf on a proper variety is ample on its base locus, its sufficiently high powers are globally generated. The special case where the base locus is finite was proved by Zariski (1962), whence the name.

The relative Fujita-Zariski theorem

TL;DR

The paper proves a relative version of the Fujita–Zariski theorem: if is an invertible sheaf on a proper whose restriction to the base locus is ample, then is globally generated for sufficiently large. It develops a relative framework using graded cohomology modules and -modules (graded -submodules) to analyze generation properties, and proves a key finiteness result via a dévissage argument and a fat-subset general-position technique. The proof proceeds by noetherian induction on , selecting general elements of to control base loci and using exact sequences to lift generation from subschemes to . Together, these methods establish a robust relative base-point freeness criterion and extend the classical result beyond the base-field setting, with potential applicability to algebraic spaces.

Abstract

We prove, with no claim to originality, a relative version of the Fujita-Zariski theorem. When the base is a field, this result is due to Fujita (1983) and states that if an invertible sheaf on a proper variety is ample on its base locus, its sufficiently high powers are globally generated. The special case where the base locus is finite was proved by Zariski (1962), whence the name.

Paper Structure

This paper contains 6 sections, 6 theorems, 8 equations.

Table of Contents

  1. Definitions, notation, and statement
  2. Graded rings and modules
  3. Fat subsets in modules, general position results
  4. Proof of the theorem
  5. A finiteness result
  6. Proof of Theorem \ref{['ThFZ']}

Key Result

Theorem 1.2

Let $\Lambda$ be a linear system on $X$ as above. Assume that the restriction of ${\mathscr{L}}$ to $B$ is ample. Then there exists an integer $t_{0}$ such that ${\mathscr{L}}^{\otimes t}$ is globally generated for all $t\geq t_{0}$.

Theorems & Definitions (10)

  • Definition 1.1
  • Theorem 1.2
  • Definition 1.4.1
  • Remark 1.4.2
  • Definition 1.5.1
  • Lemma 1.5.2
  • Proposition 1.5.3
  • Corollary 1.5.4
  • Proposition 2.1.1
  • Lemma 2.1.2