K-polystable toric Fano varieties with small alpha invariants

Abstract

For every $n\geq 2$, we exhibit an $n$-dimensional K-polystable toric $\mathbb{Q}$-Fano variety $X_n$, defined by the face fan of an explicit lattice polytope, and whose alpha invariant is exactly $\tfrac{2}{2n+1}$. This answers a question of Liu and Zhuang whether there exists an $n$-dimensional K-semistable $\mathbb{Q}$-Fano variety whose alpha invariant is between $\tfrac{1}{n+1}$ and $\tfrac1n$. The main result of this paper was obtained by Chatgpt 5.5 pro, and the Danus system based on the Rethlas system.

Disclosure

“ring maxu∈Pn , v∈Vert(Qn ) ⟨u, v⟩ = 2n−1 2 , fed into the toric formula for the alpha invariant, gives (1.1) (Section 4). Remark 1.6. The sketch of the proof of the main result of this paper was obtained by Chatgpt 5.5 pro, and later summed up, verified, and properly written by the Danus system, a specialized agent built on Rethlas and substantially more capable for fundamental mathematical research based on the Rethlas system. Human verification and”

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Classification

Substantial proof generation

Multiplier

10

Verified

2026-08-24T00:28:51.963Z

Structural counts