Improved Bounds on Ultra-Log Concavity of the Grothendieck Class of $\overline{\mathcal{M}_{0,n}}$
Fuente:
arXiv
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| Formato: | Preprint |
| Publicado: |
2025
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| _version_ | 1866916950003679232 |
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| author | Nascimento, Eduardo |
| author_facet | Nascimento, Eduardo |
| contents | The class of the fine moduli space of stable $n$-pointed curves of genus zero, $\overline{\mathcal{M}_{0,n}}$, in the Grothendieck ring of varieties encodes its Poincaré polynomial. Aluffi-Chen-Marcolli conjecture that the Grothendieck class of $\overline{\mathcal{M}_{0,n}}$ is real-rooted (and hence ultra-log-concave), and they proved an asymptotic ultra-log-concavity result for these polynomials. We build upon their work, by providing effectively computable bounds for the error term in their asymptotic formula for $\mathrm{rk}\, H^{2l}(\overline{\mathcal{M}_{0,n}})$. As a consequence, we prove that in the range $l \le \frac{n}{10\log n}$, the ultra-log-concavity inequality \[\left(\frac{\mathrm{rk}\, H^{2(l-1)}(\overline{\mathcal{M}_{0,n}})}{\binom{n-3}{l-1}}\right)^2 \ge \frac{\mathrm{rk}\, H^{2(l-2)}(\overline{\mathcal{M}_{0,n}})\mathrm{rk}\, H^{2l}(\overline{\mathcal{M}_{0,n}})}{\binom{n-3}{l-2}\binom{n-3}{l}} \]
holds for $n$ sufficiently large. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2509_11805 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Improved Bounds on Ultra-Log Concavity of the Grothendieck Class of $\overline{\mathcal{M}_{0,n}}$ Nascimento, Eduardo Algebraic Geometry Combinatorics The class of the fine moduli space of stable $n$-pointed curves of genus zero, $\overline{\mathcal{M}_{0,n}}$, in the Grothendieck ring of varieties encodes its Poincaré polynomial. Aluffi-Chen-Marcolli conjecture that the Grothendieck class of $\overline{\mathcal{M}_{0,n}}$ is real-rooted (and hence ultra-log-concave), and they proved an asymptotic ultra-log-concavity result for these polynomials. We build upon their work, by providing effectively computable bounds for the error term in their asymptotic formula for $\mathrm{rk}\, H^{2l}(\overline{\mathcal{M}_{0,n}})$. As a consequence, we prove that in the range $l \le \frac{n}{10\log n}$, the ultra-log-concavity inequality \[\left(\frac{\mathrm{rk}\, H^{2(l-1)}(\overline{\mathcal{M}_{0,n}})}{\binom{n-3}{l-1}}\right)^2 \ge \frac{\mathrm{rk}\, H^{2(l-2)}(\overline{\mathcal{M}_{0,n}})\mathrm{rk}\, H^{2l}(\overline{\mathcal{M}_{0,n}})}{\binom{n-3}{l-2}\binom{n-3}{l}} \] holds for $n$ sufficiently large. |
| title | Improved Bounds on Ultra-Log Concavity of the Grothendieck Class of $\overline{\mathcal{M}_{0,n}}$ |
| topic | Algebraic Geometry Combinatorics |
| url | https://arxiv.org/abs/2509.11805 |