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A generalized non-vanishing theorem on surfaces

Jihao Liu, Lingyao Xie

Abstract

We show that the anti-canonical bundle of any $\mathbb Q$-factorial surface is numerically effective if and only if it is pseudo-effective. To prove this, we establish a numerical non-vanishing theorem for surfaces polarized with pseudo-effective divisors. The latter answers a question of C. Fontanari.

A generalized non-vanishing theorem on surfaces

Abstract

We show that the anti-canonical bundle of any -factorial surface is numerically effective if and only if it is pseudo-effective. To prove this, we establish a numerical non-vanishing theorem for surfaces polarized with pseudo-effective divisors. The latter answers a question of C. Fontanari.

Paper Structure

This paper contains 4 sections, 9 theorems, 23 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Induction approach to the main theorem
  4. Proof of the main theorems

Key Result

Theorem 1.2

Let $X$ be a $\mathbb{Q}$-factorial projective surface and $L$ a pseudo-effective $\mathbb{Q}$-divisor on $X$ such that $K_X+L$ is pseudo-effective. Then $K_X+L$ is numerically effective.

Theorems & Definitions (22)

  • Theorem 1.2
  • Theorem 1.3
  • Definition 2.1
  • Lemma 2.2
  • proof
  • Definition 2.3
  • Definition 2.4
  • Theorem 2.5: Cone theorem for not necessarily lc pairs, cf. Amb03, Fuj17
  • Theorem 2.6
  • proof
  • ...and 12 more