Effective non-vanishing for weighted complete intersections of low codimension

TL;DR

This paper proves that certain ample divisors on low-codimension weighted complete intersections have nontrivial global sections, generalizing a numerical conjecture and linking geometric properties to numerical semigroup invariants.

Contribution

It extends the non-vanishing theorem to weighted complete intersections of codimension up to 3 by generalizing a conjecture relating global sections to Frobenius numbers.

Findings

Non-vanishing of global sections for specific divisors on weighted complete intersections.

A generalized numerical conjecture connecting algebraic geometry and numerical semigroups.

Validation of the conjecture for low codimension cases.

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.