Heights of complete intersections in toric varieties
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arXiv
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| Main Authors: | , |
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| Format: | Preprint |
| Published: |
2024
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| _version_ | 1866915074924347392 |
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| author | Gualdi, Roberto Sombra, Martín |
| author_facet | Gualdi, Roberto Sombra, Martín |
| contents | The height of a toric variety and that of its hypersurfaces can be expressed in convex-analytic terms as an adelic sum of mixed integrals of their roof functions and duals of their Ronkin functions.
Here we extend these results to the $2$-codimensional situation by presenting a limit formula predicting the typical height of the intersection of two hypersurfaces on a toric variety. More precisely, we prove that the height of the intersection cycle of two effective divisors translated by a strict sequence of torsion points converges to an adelic sum of mixed integrals of roof and duals of Ronkin functions.
This partially confirms a previous conjecture of the authors about the average height of families of complete intersections in toric varieties. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2412_16308 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Heights of complete intersections in toric varieties Gualdi, Roberto Sombra, Martín Algebraic Geometry Number Theory Primary 14G40, Secondary 11G50, 14M25, 52A39 The height of a toric variety and that of its hypersurfaces can be expressed in convex-analytic terms as an adelic sum of mixed integrals of their roof functions and duals of their Ronkin functions. Here we extend these results to the $2$-codimensional situation by presenting a limit formula predicting the typical height of the intersection of two hypersurfaces on a toric variety. More precisely, we prove that the height of the intersection cycle of two effective divisors translated by a strict sequence of torsion points converges to an adelic sum of mixed integrals of roof and duals of Ronkin functions. This partially confirms a previous conjecture of the authors about the average height of families of complete intersections in toric varieties. |
| title | Heights of complete intersections in toric varieties |
| topic | Algebraic Geometry Number Theory Primary 14G40, Secondary 11G50, 14M25, 52A39 |
| url | https://arxiv.org/abs/2412.16308 |