Toric Representation Type of the Veronese Surface

Abstract

In this article we determine the toric representation type of the Veronese surface $(\mathbb{P}^2,\mathcal{O}_{\mathbb{P}^2}(d))$. Based on Klyachko filtrations, we introduce an explicit criterion for a toric vector bundle of arbitrary rank to be arithmetically Cohen--Macaulay. For $d \geq 3$, suitable configurations of partial flags produce stable toric $d$-aCM bundles corresponding to imaginary non-isotropic Schur roots of star-shaped quivers. Their self-extensions give an exact representation embedding of $\operatorname{mod}\mathbb{C}\langle x,y\rangle$, proving that the Veronese surface is toric-wild precisely for $d \geq 3$, while it is toric-finite for $d=1,2$. For $d=3,4$, suitable twists of the basic stable bundles are Ulrich, and the same construction proves that the corresponding Veronese surfaces are toric Ulrich-wild.