On the blow-up formula of the Chow weights for polarized toric manifolds

Let $X$ be a smooth projective toric variety, and let $\widetilde{X}$ denote the blow-up of $X$ at finitely many distinct torus-invariant points. In this paper, we derive an explicit combinatorial formula for the Chow weight of $\widetilde{X}$ in terms of the underlying toric manifold $X$ and the symplectic cuts of its associated Delzant polytope. As an application, we study toric blow-ups of the projective plane and compare their Chow stability with that of blow-ups at general points.