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

Abstract

Let $X$ be a smooth projective toric variety, and let $\tilde{X}$ 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 $\tilde{X}$ 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.

Third version (21 pages): We have stated the precise relationship between the toric Chow weights used in this paper and the general algebro-geometric notion of Chow stability. In addition, we have clarified that Proposition 2.3 applies to certain specialized cases