Veronese曲面的环面表示型
Toric Representation Type of the Veronese Surface
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中文总结 AI 辅助
本文确定Veronese曲面的环面表示型，基于Klyachko滤子引入aCM环面向量丛的判据，证明$d\geq3$时其为环面野型，$d=1,2$时为环面有限型，$d=3,4$时为环面Ulrich野型。
中文摘要 AI 辅助
本文确定了Veronese曲面$(\text{P}^2,\mathcal{O}_{\text{P}^2}(d))$的环面表示型。基于Klyachko滤子，我们为任意秩的环面向量丛引入了一个显式判据，以判定其是否为算术Cohen-Macaulay（aCM）。当$d\geq3$时，合适的部分旗配置会生成稳定的环面$d$-aCM丛，对应星形箭图的虚非迷向Schur根；这类丛的自扩张给出了$\text{mod}\mathbb{C}\langle x,y\rangle$的精确表示嵌入，证明了Veronese曲面恰在$d\geq3$时为环面野型，而在$d=1,2$时为环面有限型。对于$d=3,4$，基本稳定丛的合适扭曲为Ulrich丛，且相同构造证明对应Veronese曲面为环面Ulrich野型。
英文摘要
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.
