Effective nonvanishing for weighted complete intersections of codimension two

TL;DR

This paper proves Kawamata's effective nonvanishing conjecture for a specific class of weighted complete intersections, showing that certain linear systems are nonempty under given ampleness conditions.

Contribution

It establishes the conjecture for quasismooth weighted complete intersections of codimension two, a case not previously confirmed.

Findings

Kawamata's conjecture holds for these intersections.

The linear system |H| is nonempty under specified conditions.

Provides new evidence supporting the conjecture in weighted settings.

Abstract

We show Kawamata's effective nonvanishing conjecture (also known as the Ambro--Kawamata nonvanishing conjecture) holds for quasismooth weighted complete intersections of codimension $2$. Namely, for a quasismooth weighted complete intersection $X$ of codimension $2$ and an ample Cartier divisor $H$ on $X$ such that $H-K_X$ is ample, the linear system $|H|$ is nonempty.