Rational Complexity-One $\boldsymbol{T}$-Varieties Are Well-Poised
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Bibliographic record
Abstract
Abstract Given an affine rational complexity-one $T$-variety $X$, we construct an explicit embedding of $X$ in affine space ${\mathbb{A}}^n$. We show that this embedding is well-poised, that is, every initial ideal of $I_X$ is a prime ideal, and we determine the tropicalization ${\mathrm{Trop}}(X^\circ)$. We then study valuations of the coordinate ring $R_X$ of $X$ which respect the torus action, showing that for full rank valuations, the natural generators of $R_X$ form a Khovanskii basis. This allows us to determine Newton–Okounkov bodies of rational projective complexity-one $T$-varieties, partially recovering (and generalizing) results of Petersen. We apply our results to describe all integral special fibres of ${\mathbb{K}}^*\times T$-equivariant degenerations of rational projective complexity-one $T$-varieties, generalizing a result of Süß and Ilten.
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Full frame distilled prediction
Teacher imitationNot calibrated prevalence, not ground truth. Human validation pending. Learned from the 10,348 direct Codex labels and 10,348 direct Gemma labels. Candidate is the union of thresholded teacher heads; consensus is their intersection. These outputs are machine_predicted_unvalidated and are not human labels or direct frontier model labels.
Codex and Gemma teacher scores by category
| Category | Codex | Gemma |
|---|---|---|
| Metaresearch | 0.001 | 0.001 |
| Meta-epidemiology (narrow) | 0.000 | 0.000 |
| Meta-epidemiology (broad) | 0.000 | 0.000 |
| Bibliometrics | 0.000 | 0.000 |
| Science and technology studies | 0.001 | 0.000 |
| Scholarly communication | 0.003 | 0.001 |
| Open science | 0.004 | 0.001 |
| Research integrity | 0.000 | 0.000 |
| Insufficient payload (model declined to judge) | 0.000 | 0.001 |
Machine scores (provisional)
The two teacher heads of the student model, read on this work. A score orders the frame for review; it never asserts a category, and the validation status ships verbatim with every row.
Baseline scores from an immature model (maturity gate not passed, 7 training rounds). Scores rank; they never assert a category.
score_only:v0-immature-baseline · verbatim from the scoring run: score_only means the number may rank works, and no category label ships from it