Kapranov degrees
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arXiv
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| Main Authors: | , , , |
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| Format: | Preprint |
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2023
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| _version_ | 1866908533579055104 |
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| author | Brakensiek, Joshua Eur, Christopher Larson, Matt Li, Shiyue |
| author_facet | Brakensiek, Joshua Eur, Christopher Larson, Matt Li, Shiyue |
| contents | The moduli space of stable rational curves with marked points has two distinguished families of maps: the forgetful maps, given by forgetting some of the markings, and the Kapranov maps, given by complete linear series of $ψ$-classes. The collection of all these maps embeds the moduli space into a product of projective spaces. We call the multidegrees of this embedding ``Kapranov degrees,'' which include as special cases the work of Witten, Silversmith, Gallet--Grasegger--Schicho, Castravet--Tevelev, Postnikov, Cavalieri--Gillespie--Monin, and Gillespie--Griffins--Levinson. We establish, in terms of a combinatorial matching condition, upper bounds for Kapranov degrees and a characterization of their positivity. The positivity characterization answers a question of Silversmith and gives a new proof of Laman's theorem characterizing generically rigid graphs in the plane. We achieve this by proving a recursive formula for Kapranov degrees and by using tools from the theory of error correcting codes. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2308_12285 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Kapranov degrees Brakensiek, Joshua Eur, Christopher Larson, Matt Li, Shiyue Algebraic Geometry Combinatorics The moduli space of stable rational curves with marked points has two distinguished families of maps: the forgetful maps, given by forgetting some of the markings, and the Kapranov maps, given by complete linear series of $ψ$-classes. The collection of all these maps embeds the moduli space into a product of projective spaces. We call the multidegrees of this embedding ``Kapranov degrees,'' which include as special cases the work of Witten, Silversmith, Gallet--Grasegger--Schicho, Castravet--Tevelev, Postnikov, Cavalieri--Gillespie--Monin, and Gillespie--Griffins--Levinson. We establish, in terms of a combinatorial matching condition, upper bounds for Kapranov degrees and a characterization of their positivity. The positivity characterization answers a question of Silversmith and gives a new proof of Laman's theorem characterizing generically rigid graphs in the plane. We achieve this by proving a recursive formula for Kapranov degrees and by using tools from the theory of error correcting codes. |
| title | Kapranov degrees |
| topic | Algebraic Geometry Combinatorics |
| url | https://arxiv.org/abs/2308.12285 |