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We describe the parameter test submodule of $S/J$ in terms of the test ideal of the pair $(R, I)$ when a reduction of $I$ is a complete intersection or almost complete intersection. As an application, we deduce a criterion for when $S/J$ has $F$-rational singularities in these cases. We also compare the $F$-pure threshold of $(R, I)$ and $(S, J)$."},"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":"1607.03972","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AC","submitted_at":"2016-07-14T00:58:57Z","cross_cats_sorted":["math.AG"],"title_canon_sha256":"227e85278d6184c63d002e27400c52af03e55179ab3e05c54990e6c853e0d2e7","abstract_canon_sha256":"6b1e5f1280826267193dd3460b1fa02ce0e0e9c54e42542d393241f6cdd7880e"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T00:20:49.346550Z","signature_b64":"fYxfjb0a+G1Dh9ZYkQQ4Q26fqZb/rednDbR17fw10O7m/lD/qi6nR9dl+V+TdnHKajBS8+hLlxsGLt2NzL8WAw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"5df48ebe0732af56139be930605be77b10ff9377123c17ec7d65220f1bb163b7","last_reissued_at":"2026-05-18T00:20:49.345762Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T00:20:49.345762Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"$F$-singularities under generic linkage","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.AG"],"primary_cat":"math.AC","authors_text":"Janet Page, Linquan Ma, Rebecca R.G., Wenliang Zhang, William Taylor","submitted_at":"2016-07-14T00:58:57Z","abstract_excerpt":"Let $R=k[x_1,\\dots,x_n]$ be a polynomial ring over a prefect field of positive characteristic. Let $I$ be an unmixed ideal in $R$ and let $J$ be a generic link of $I$ in $S=R[u_{ij}]_{c \\times r}$. We describe the parameter test submodule of $S/J$ in terms of the test ideal of the pair $(R, I)$ when a reduction of $I$ is a complete intersection or almost complete intersection. As an application, we deduce a criterion for when $S/J$ has $F$-rational singularities in these cases. 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