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Effective non-vanishing for weighted complete intersections of low codimension

Alessandro Passantino

TL;DR

The paper addresses the non-vanishing problem in the Ambro-Kawamata framework for quasi-smooth weighted complete intersections by translating geometry into a numerical Frobenius-type problem on $h$-regular pairs. It proves that effective non-vanishing holds for WCIs of codimension at most $3$ not forming a linear cone, by establishing a lower bound $\delta(d;a)\ge F^h(a)$ for regular pairs and using a suite of reductions to the regular case. Recursive Frobenius bounds and degree- and weight- gcd reductions are developed to control the semigroup bounds that govern sections, yielding equivalence of Frobenius-type conjectures in the small-codimension regime. The results connect geometric existence of global sections to classical numerical semigroup theory, providing a practical numerical toolkit for broader investigations in higher codimension and general type.

Abstract

We show that on quasi-smooth weighted complete intersections of codimension at most 3, any ample Cartier divisor $H$ such that $H-K_X$ is ample admits a nontrivial global section. This is done by proving a generalisation of a numerical conjecture formulated by Pizzato, Sano and Tasin, which relates the existence of global sections of $H$ to the Frobenius number of the numerical semigroup generated by the weights of the ambient projective space.

Effective non-vanishing for weighted complete intersections of low codimension

TL;DR

The paper addresses the non-vanishing problem in the Ambro-Kawamata framework for quasi-smooth weighted complete intersections by translating geometry into a numerical Frobenius-type problem on -regular pairs. It proves that effective non-vanishing holds for WCIs of codimension at most not forming a linear cone, by establishing a lower bound for regular pairs and using a suite of reductions to the regular case. Recursive Frobenius bounds and degree- and weight- gcd reductions are developed to control the semigroup bounds that govern sections, yielding equivalence of Frobenius-type conjectures in the small-codimension regime. The results connect geometric existence of global sections to classical numerical semigroup theory, providing a practical numerical toolkit for broader investigations in higher codimension and general type.

Abstract

We show that on quasi-smooth weighted complete intersections of codimension at most 3, any ample Cartier divisor such that is ample admits a nontrivial global section. This is done by proving a generalisation of a numerical conjecture formulated by Pizzato, Sano and Tasin, which relates the existence of global sections of to the Frobenius number of the numerical semigroup generated by the weights of the ambient projective space.
Paper Structure (14 sections, 30 theorems, 49 equations)

This paper contains 14 sections, 30 theorems, 49 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Weighted projective spaces
  4. Weighted complete intersections
  5. $h$-regular pairs
  6. Effective non-vanishing and the Frobenius problem
  7. About the Frobenius problem
  8. Properties of $h$-regular pairs and Frobenius numbers
  9. Properties of Frobenius numbers
  10. Reduction to degrees with no common factors
  11. Recursive bounds on Frobenius numbers
  12. Reduction to the regular case
  13. Effective non-vanishing
  14. About the codimension of minimal pairs

Key Result

Theorem 1.2

Let $X \subset \mathbb{P}(a_0,\ldots,a_n)$ be a well-formed quasi-smooth weighted complete intersection which is not a linear cone, such that $\mathrm{codim}X \leq 3$. Then, for any ample Cartier divisor $H$ on $X$ such that $H - K_X$ is ample, $H^0(X, H) \neq 0$.

Theorems & Definitions (60)

  • Conjecture 1.1: Ambro-Kawamata; kaw00
  • Theorem 1.2: =Theorem \ref{['thm:main']}
  • Conjecture 1.3: tasin; Conjecture \ref{['conjecture:frob']}
  • Theorem 1.4: =Corollary \ref{['cor:maincor']}
  • Theorem 1.5: =Corollary \ref{['cor:noasterisk']}
  • Definition 2.1
  • Definition 2.2
  • Remark 2.3
  • Definition 2.4
  • Definition 2.5: dimca, Definition 1
  • ...and 50 more