Examples of singular toric varieties with certain numerical conditions

TL;DR

This paper provides examples of specific singular toric varieties with positive sums of squared divisors and characterizes the effective 2-cycle cone for certain Picard number cases, advancing understanding of their geometric properties.

Contribution

It introduces new examples of Q-factorial projective toric varieties with positive divisor sums and explicitly describes the effective 2-cycle cone for Picard number two varieties.

Findings

Examples of Q-factorial projective toric varieties with positive divisor sums.

Explicit generators for the cone of effective 2-cycles in certain toric varieties.

Enhanced understanding of the geometry of singular toric varieties.

Abstract

We give various examples of Q-factorial projective toric varieties such that the sum of the squared torus invariant prime divisors is positive. We also determine the generators for the cone of effective $2$-cycles on a toric variety of Picard number two. This result is convenient to explain our examples.