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| Format: | Preprint |
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2026
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| Accès en ligne: | https://arxiv.org/abs/2604.14536 |
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| _version_ | 1866910133350563840 |
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| author | Debnath, Arkamouli Zeng, Michael Ruofan |
| author_facet | Debnath, Arkamouli Zeng, Michael Ruofan |
| contents | Oriented cohomology theories provide a general framework to perform intersection-theory-type calculus. The Chow ring, algebraic $K$-theory, and Levine--Morel's algebraic cobordism are all instances of such theories satisfying $\mathbb A^1$-invariance. Topological Hochschild homology, topological cyclic homology, and Hodge cohomology are important examples of theories without $\mathbb A^1$-invariance. In this paper, we prove an additive blowup formula for oriented cohomology theories in the non-$\mathbb A^1$-invariant category of motivic spectra, developed by Annala, Hoyois, and Iwasa. Then, we specialize to $\mathbb A^1$-invariant theories and give presentations of oriented cohomology rings of the blowup of a smooth scheme along a smooth center. We compute explicit examples of such presentations for the cases of del Pezzo surfaces, the blowup of $\mathbb P^3$ along the twisted cubic, and the blowup of $\mathbb P^5$ along the Veronese surface, which can be identified with the moduli space of complete conics. We demonstrate that one can recover solutions to classical enumerative geometry problems, such as Steiner's $3264$ conics, using arbitrary oriented cohomology theories. Finally, we give a presentation of oriented cohomology rings of $\overline M_{0,n}$, which generalizes Keel's presentation of the Chow ring. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2604_14536 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Oriented Cohomology Rings of Some Moduli Spaces via Blowups Debnath, Arkamouli Zeng, Michael Ruofan Algebraic Geometry Algebraic Topology 14C17, 14F42, 14H10, 14N10 Oriented cohomology theories provide a general framework to perform intersection-theory-type calculus. The Chow ring, algebraic $K$-theory, and Levine--Morel's algebraic cobordism are all instances of such theories satisfying $\mathbb A^1$-invariance. Topological Hochschild homology, topological cyclic homology, and Hodge cohomology are important examples of theories without $\mathbb A^1$-invariance. In this paper, we prove an additive blowup formula for oriented cohomology theories in the non-$\mathbb A^1$-invariant category of motivic spectra, developed by Annala, Hoyois, and Iwasa. Then, we specialize to $\mathbb A^1$-invariant theories and give presentations of oriented cohomology rings of the blowup of a smooth scheme along a smooth center. We compute explicit examples of such presentations for the cases of del Pezzo surfaces, the blowup of $\mathbb P^3$ along the twisted cubic, and the blowup of $\mathbb P^5$ along the Veronese surface, which can be identified with the moduli space of complete conics. We demonstrate that one can recover solutions to classical enumerative geometry problems, such as Steiner's $3264$ conics, using arbitrary oriented cohomology theories. Finally, we give a presentation of oriented cohomology rings of $\overline M_{0,n}$, which generalizes Keel's presentation of the Chow ring. |
| title | Oriented Cohomology Rings of Some Moduli Spaces via Blowups |
| topic | Algebraic Geometry Algebraic Topology 14C17, 14F42, 14H10, 14N10 |
| url | https://arxiv.org/abs/2604.14536 |