On Property $N_p$ of line bundles on smooth projective toric varieties

We establish a criterion for Property $N_p$ for line bundles on a class of smooth projective toric varieties.

More precisely, we prove that if a smooth projective toric variety $X$ of dimension $n\ge2$ satisfies the uniform unimodularity condition and the Thomsen stratification intersection-number condition, then any line bundle $L$ on $X$ with $L\cdot C\ge n-1+p$ for every $T$-invariant curve $C$ satisfies Property $N_p$.

We also show that these two conditions hold for several families of toric varieties and are preserved under finite products.