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We prove that the tautological ring of $M_{g,n}^{rt}$ has Poincar\\'e duality if and only if the same holds for the tautological ring of $C_g^n$. We also obtain a presentation of the tautological ring of $M_{g,n}^{rt}$ as an algebra over the tautological ring of $C_g^n$. This proves a conjecture of Tavakol. Our results are valid in the more general setting of wonderful compactifications."},"verification_status":{"content_addressed":true,"pith_receipt":true,"author_attested":false,"weak_author_claims":0,"strong_author_claims":0,"externally_anchored":false,"storage_verified":false,"citation_signatures":0,"replication_records":0,"graph_snapshot":true,"references_resolved":false,"formal_links_present":false},"canonical_record":{"source":{"id":"1501.04742","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AG","submitted_at":"2015-01-20T09:11:42Z","cross_cats_sorted":[],"title_canon_sha256":"da08e9a8f9e1260e0c44f6bd5894677e973ad8ec4c6d681cfcc49f610232c2fe","abstract_canon_sha256":"c50fcd288a850d4e6cfe5820d4a65056272475684cac60ff589edccd0bda29a5"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T01:04:33.451554Z","signature_b64":"niUQOjGkZJzx/JSOu4ei+vNTZCC22qAVzGbl1wrjrh+m2cUP+Iat8VV6ij5htWrsmhEsoE6McKflNZ2qSXDcCA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"40d02ae7c3e7e8d4578fa9741710ccc64c07373e0cddcf54ce8cbd946b231afd","last_reissued_at":"2026-05-18T01:04:33.450813Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T01:04:33.450813Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Poincar\\'e duality of wonderful compactifications and tautological rings","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.AG","authors_text":"Dan Petersen","submitted_at":"2015-01-20T09:11:42Z","abstract_excerpt":"Let $g \\geq 2$. Let $M_{g,n}^{rt}$ be the moduli space of $n$-pointed genus $g$ curves with rational tails. Let $C_g^n$ be the $n$-fold fibered power of the universal curve over $M_g$. We prove that the tautological ring of $M_{g,n}^{rt}$ has Poincar\\'e duality if and only if the same holds for the tautological ring of $C_g^n$. We also obtain a presentation of the tautological ring of $M_{g,n}^{rt}$ as an algebra over the tautological ring of $C_g^n$. This proves a conjecture of Tavakol. 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