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We find relations between these invariants and also give necessary and sufficient conditions for a $1$-parameter family to be Whitney equisingular. As an application, we show that a family $(X_t,0)$ is Zariski equisingular if and only if it is Whitney equisingular and the numbers of cusps and double folds of a generic linear projection are constant on $t$."},"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":"1507.01483","kind":"arxiv","version":3},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AG","submitted_at":"2015-07-06T14:47:43Z","cross_cats_sorted":[],"title_canon_sha256":"e7b7a0cde98805dc1f445e8869f06e51d2fb09f04b71ad2696077535d85c539d","abstract_canon_sha256":"4075c34664901db5bbf06f38cfced084dba9816f771e96c63960b8b24aab1d72"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T01:12:49.831424Z","signature_b64":"nM+2TH/6bICq77/qqsKcFFaNjnYKbLzXTyqjzlCriHW4E6qDP7SswQUBpQNcQn+LKPXXsH+4ffvO3W42+NcbDA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"27687d362fddae12ccc4c2a3179f4cd806b02b3682b0cc06f1fba891200dd019","last_reissued_at":"2026-05-18T01:12:49.830936Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T01:12:49.830936Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Equisingularity of map germs from a surface to the plane","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.AG","authors_text":"B. 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