# Effective non-vanishing for Fano weighted complete intersections

## TL;DR

This paper proves the Ambro-Kawamata non-vanishing conjecture for certain Fano and Calabi-Yau weighted complete intersections, establishing the existence of sections for ample divisors and smoothness of general elements.

## Contribution

It extends non-vanishing results to quasi-smooth weighted complete intersections and verifies related conjectures for weighted hypersurfaces, introducing the concept of regular pairs.

## Key findings

- Non-vanishing holds for quasi-smooth Fano or Calabi-Yau weighted complete intersections.
- Verifies Ambro-Kawamata's conjecture for quasi-smooth weighted hypersurfaces.
- Confirms Fujita's freeness conjecture for Gorenstein quasi-smooth weighted hypersurfaces.

## Abstract

We show that Ambro-Kawamata's non-vanishing conjecture holds true for a quasi-smooth WCI X which is Fano or Calabi-Yau, i.e. we prove that, if H is an ample Cartier divisor on X, then |H| is not empty. If X is smooth, we further show that the general element of |H| is smooth. We then verify Ambro-Kawamata's conjecture for any quasi-smooth weighted hypersurface. We also verify Fujita's freeness conjecture for a Gorenstein quasi-smooth weighted hypersurface. For the proofs, we introduce the arithmetic notion of regular pairs and enlighten some interesting connection with the Frobenius coin problem.