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| Main Authors: | , |
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| Format: | Preprint |
| Published: |
2021
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2112.02847 |
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| _version_ | 1866913361846861824 |
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| author | Jiang, Chen Li, Zhiyuan |
| author_facet | Jiang, Chen Li, Zhiyuan |
| contents | Let $X$ be a projective variety of dimension $n$ over an algebraically closed field of arbitrary characteristic and let $A, B, C$ be nef divisors on $X$. We show that for any integer $1\leq k\leq n-1$, $$ (B^k\cdot A^{n-k})\cdot (A^k\cdot C^{n-k})\geq \frac{k!(n-k)!}{n!}(A^n)\cdot (B^k\cdot C^{n-k}). $$ The same inequality in the analytic setting was obtained by Lehmann and Xiao for compact Kähler manifolds using the Calabi--Yau theorem, while our approach is purely algebraic using (multipoint) Okounkov bodies. We also discuss applications of this inequality to Bézout-type inequalities and inequalities on degrees of dominant rational self-maps. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2112_02847 |
| institution | arXiv |
| publishDate | 2021 |
| record_format | arxiv |
| spellingShingle | Algebraic reverse Khovanskii--Teissier inequality via Okounkov bodies Jiang, Chen Li, Zhiyuan Algebraic Geometry Let $X$ be a projective variety of dimension $n$ over an algebraically closed field of arbitrary characteristic and let $A, B, C$ be nef divisors on $X$. We show that for any integer $1\leq k\leq n-1$, $$ (B^k\cdot A^{n-k})\cdot (A^k\cdot C^{n-k})\geq \frac{k!(n-k)!}{n!}(A^n)\cdot (B^k\cdot C^{n-k}). $$ The same inequality in the analytic setting was obtained by Lehmann and Xiao for compact Kähler manifolds using the Calabi--Yau theorem, while our approach is purely algebraic using (multipoint) Okounkov bodies. We also discuss applications of this inequality to Bézout-type inequalities and inequalities on degrees of dominant rational self-maps. |
| title | Algebraic reverse Khovanskii--Teissier inequality via Okounkov bodies |
| topic | Algebraic Geometry |
| url | https://arxiv.org/abs/2112.02847 |