On the tangent bundle and the divisor theory of a general matroid
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866917337026789376 |
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| author | Cheng, Ronnie |
| author_facet | Cheng, Ronnie |
| contents | Extending classical algebro-geometric constructions to arbitrary matroids, we construct a $K$-class $T_M\in K(M)$ for every loopless matroid $M$. When $M$ is realizable by a linear subspace $L$, $T_M$ recovers the $K$-class of the tangent bundle of the wonderful compactification $W_L$. We derive two formulas for the total Chern class of $T_M$ (one combinatorial and one geometric) and show that the associated Todd class agrees with the Todd class appearing in the matroid Hirzebruch--Riemann--Roch formula. To develop a positivity theory entirely at the combinatorial level, we introduce the notion of ``fake effective cone,'' a combinatorial analogue of the classical effective cone, and use it to characterize big and nef divisors in $A(M)$. Finally, we define the $β_S$ classes, obtained from Cremona conjugates of the classical $α_S$ classes, and study their properties to provide a rich and computable family of combinatorially nef divisors. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2510_06609 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | On the tangent bundle and the divisor theory of a general matroid Cheng, Ronnie Algebraic Geometry Combinatorics 14M99 (Primary) 05B35, 52C35 (Secondary) Extending classical algebro-geometric constructions to arbitrary matroids, we construct a $K$-class $T_M\in K(M)$ for every loopless matroid $M$. When $M$ is realizable by a linear subspace $L$, $T_M$ recovers the $K$-class of the tangent bundle of the wonderful compactification $W_L$. We derive two formulas for the total Chern class of $T_M$ (one combinatorial and one geometric) and show that the associated Todd class agrees with the Todd class appearing in the matroid Hirzebruch--Riemann--Roch formula. To develop a positivity theory entirely at the combinatorial level, we introduce the notion of ``fake effective cone,'' a combinatorial analogue of the classical effective cone, and use it to characterize big and nef divisors in $A(M)$. Finally, we define the $β_S$ classes, obtained from Cremona conjugates of the classical $α_S$ classes, and study their properties to provide a rich and computable family of combinatorially nef divisors. |
| title | On the tangent bundle and the divisor theory of a general matroid |
| topic | Algebraic Geometry Combinatorics 14M99 (Primary) 05B35, 52C35 (Secondary) |
| url | https://arxiv.org/abs/2510.06609 |