{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2015:LSL2GRHSOA3AJVXYTC5OTYIDXG","short_pith_number":"pith:LSL2GRHS","canonical_record":{"source":{"id":"1508.06991","kind":"arxiv","version":4},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AG","submitted_at":"2015-08-27T20:04:13Z","cross_cats_sorted":[],"title_canon_sha256":"077af3832e24434e90365ede52e6da70fafe771bbbf5e72f52afbc16281daa62","abstract_canon_sha256":"2eaa5349d854c7492efc142157bc5c9f48f86f51a87be5a50bf1aacb9c738a5e"},"schema_version":"1.0"},"canonical_sha256":"5c97a344f2703604d6f898bae9e103b9ba8a1e88577b96d853c98ccc3ab970ba","source":{"kind":"arxiv","id":"1508.06991","version":4},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1508.06991","created_at":"2026-05-17T23:59:26Z"},{"alias_kind":"arxiv_version","alias_value":"1508.06991v4","created_at":"2026-05-17T23:59:26Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1508.06991","created_at":"2026-05-17T23:59:26Z"},{"alias_kind":"pith_short_12","alias_value":"LSL2GRHSOA3A","created_at":"2026-05-18T12:29:29Z"},{"alias_kind":"pith_short_16","alias_value":"LSL2GRHSOA3AJVXY","created_at":"2026-05-18T12:29:29Z"},{"alias_kind":"pith_short_8","alias_value":"LSL2GRHS","created_at":"2026-05-18T12:29:29Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2015:LSL2GRHSOA3AJVXYTC5OTYIDXG","target":"record","payload":{"canonical_record":{"source":{"id":"1508.06991","kind":"arxiv","version":4},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AG","submitted_at":"2015-08-27T20:04:13Z","cross_cats_sorted":[],"title_canon_sha256":"077af3832e24434e90365ede52e6da70fafe771bbbf5e72f52afbc16281daa62","abstract_canon_sha256":"2eaa5349d854c7492efc142157bc5c9f48f86f51a87be5a50bf1aacb9c738a5e"},"schema_version":"1.0"},"canonical_sha256":"5c97a344f2703604d6f898bae9e103b9ba8a1e88577b96d853c98ccc3ab970ba","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-17T23:59:26.805637Z","signature_b64":"Xz+wz5Bo8mnnUJlO89wNARhpAeZXL2tncDCvfSrO1V8NSD1gNvU/bOYp6afd3iogrTDSeg9I5uIADjpc/cFhBw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"5c97a344f2703604d6f898bae9e103b9ba8a1e88577b96d853c98ccc3ab970ba","last_reissued_at":"2026-05-17T23:59:26.805140Z","signature_status":"signed_v1","first_computed_at":"2026-05-17T23:59:26.805140Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1508.06991","source_version":4,"attestation_state":"computed"},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-17T23:59:26Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"wZ+fZJqCKhRcurQP5IF3z/6RfuzuzaIdOsu/FrnWnbWGs6PIx9K19ZgHWdtLWRN5HRH46hGo2yXXgSNRPLHJCA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-04T21:28:07.901990Z"},"content_sha256":"52f4ddeb5687d397b2a06ea2c3974252efb9c41c98b03aa7bfe35df01bc29d51","schema_version":"1.0","event_id":"sha256:52f4ddeb5687d397b2a06ea2c3974252efb9c41c98b03aa7bfe35df01bc29d51"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2015:LSL2GRHSOA3AJVXYTC5OTYIDXG","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"GIT semistability of Hilbert points of Milnor algebras","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.AG","authors_text":"Maksym Fedorchuk","submitted_at":"2015-08-27T20:04:13Z","abstract_excerpt":"Our first result is that a homogeneous form $F$ in $n$ variables is GIT semistable with respect to the natural $SL(n)$-action if and only if the first non-trivial Hilbert point of the associated Milnor algebra is semistable. We also prove that the induced morphism on the GIT quotients is finite, and injective on the locus of stable forms. Our second result is that the associated form of $F$, also known as the Macaulay inverse system of the Milnor algebra of $F$, and which is apolar to the last non-trivial Hilbert point of the Milnor algebra, is GIT semistable whenever $F$ is a smooth form. The"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1508.06991","kind":"arxiv","version":4},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"references":{"count":0,"sample":[],"resolved_work":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","internal_anchors":0},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"},"verdict_id":null},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-17T23:59:26Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"Xmv/X0yzlpSwbzW4/+CHvNCoxuSvnrniNXqBvMQdAKhsGOMibPoKvJ/8jCefZb8PowKUuQ8XFLSg03viFwuRBw==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-04T21:28:07.902346Z"},"content_sha256":"a9cb7abf2ae8aab7c1c1bfbd9d87135fb65ddef3d1568b65b6b4067edf582c96","schema_version":"1.0","event_id":"sha256:a9cb7abf2ae8aab7c1c1bfbd9d87135fb65ddef3d1568b65b6b4067edf582c96"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/LSL2GRHSOA3AJVXYTC5OTYIDXG/bundle.json","state_url":"https://pith.science/pith/LSL2GRHSOA3AJVXYTC5OTYIDXG/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/LSL2GRHSOA3AJVXYTC5OTYIDXG/bundle.json","status":"primary"}],"public_keys":[{"key_id":"pith-v1-2026-05","algorithm":"ed25519","format":"raw","public_key_b64":"stVStoiQhXFxp4s2pdzPNoqVNBMojDU/fJ2db5S3CbM=","public_key_hex":"b2d552b68890857171a78b36a5dccf368a953413288c353f7c9d9d6f94b709b3","fingerprint_sha256_b32_first128bits":"RVFV5Z2OI2J3ZUO7ERDEBCYNKS","fingerprint_sha256_hex":"8d4b5ee74e4693bcd1df2446408b0d54","rotates_at":null,"url":"https://pith.science/pith-signing-key.json","notes":"Pith uses this Ed25519 key to sign canonical record SHA-256 digests. Verify with: ed25519_verify(public_key, message=canonical_sha256_bytes, signature=base64decode(signature_b64))."}],"merge_version":"pith-open-graph-merge-v1","built_at":"2026-06-04T21:28:07Z","links":{"resolver":"https://pith.science/pith/LSL2GRHSOA3AJVXYTC5OTYIDXG","bundle":"https://pith.science/pith/LSL2GRHSOA3AJVXYTC5OTYIDXG/bundle.json","state":"https://pith.science/pith/LSL2GRHSOA3AJVXYTC5OTYIDXG/state.json","well_known_bundle":"https://pith.science/.well-known/pith/LSL2GRHSOA3AJVXYTC5OTYIDXG/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2015:LSL2GRHSOA3AJVXYTC5OTYIDXG","merge_version":"pith-open-graph-merge-v1","event_count":2,"valid_event_count":2,"invalid_event_count":0,"equivocation_count":0,"current":{"canonical_record":{"metadata":{"abstract_canon_sha256":"2eaa5349d854c7492efc142157bc5c9f48f86f51a87be5a50bf1aacb9c738a5e","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AG","submitted_at":"2015-08-27T20:04:13Z","title_canon_sha256":"077af3832e24434e90365ede52e6da70fafe771bbbf5e72f52afbc16281daa62"},"schema_version":"1.0","source":{"id":"1508.06991","kind":"arxiv","version":4}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1508.06991","created_at":"2026-05-17T23:59:26Z"},{"alias_kind":"arxiv_version","alias_value":"1508.06991v4","created_at":"2026-05-17T23:59:26Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1508.06991","created_at":"2026-05-17T23:59:26Z"},{"alias_kind":"pith_short_12","alias_value":"LSL2GRHSOA3A","created_at":"2026-05-18T12:29:29Z"},{"alias_kind":"pith_short_16","alias_value":"LSL2GRHSOA3AJVXY","created_at":"2026-05-18T12:29:29Z"},{"alias_kind":"pith_short_8","alias_value":"LSL2GRHS","created_at":"2026-05-18T12:29:29Z"}],"graph_snapshots":[{"event_id":"sha256:a9cb7abf2ae8aab7c1c1bfbd9d87135fb65ddef3d1568b65b6b4067edf582c96","target":"graph","created_at":"2026-05-17T23:59:26Z","signer":{"key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signer_id":"pith.science","signer_type":"pith_registry"},"payload":{"graph_snapshot":{"author_claims":{"count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","strong_count":0},"builder_version":"pith-number-builder-2026-05-17-v1","claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"paper":{"abstract_excerpt":"Our first result is that a homogeneous form $F$ in $n$ variables is GIT semistable with respect to the natural $SL(n)$-action if and only if the first non-trivial Hilbert point of the associated Milnor algebra is semistable. We also prove that the induced morphism on the GIT quotients is finite, and injective on the locus of stable forms. Our second result is that the associated form of $F$, also known as the Macaulay inverse system of the Milnor algebra of $F$, and which is apolar to the last non-trivial Hilbert point of the Milnor algebra, is GIT semistable whenever $F$ is a smooth form. The","authors_text":"Maksym Fedorchuk","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AG","submitted_at":"2015-08-27T20:04:13Z","title":"GIT semistability of Hilbert points of Milnor algebras"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1508.06991","kind":"arxiv","version":4},"verdict":{"created_at":null,"id":null,"model_set":{},"one_line_summary":"","pipeline_version":null,"pith_extraction_headline":"","strongest_claim":"","weakest_assumption":""}},"verdict_id":null}}],"author_attestations":[],"timestamp_anchors":[],"storage_attestations":[],"citation_signatures":[],"replication_records":[],"corrections":[],"mirror_hints":[],"record_created":{"event_id":"sha256:52f4ddeb5687d397b2a06ea2c3974252efb9c41c98b03aa7bfe35df01bc29d51","target":"record","created_at":"2026-05-17T23:59:26Z","signer":{"key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signer_id":"pith.science","signer_type":"pith_registry"},"payload":{"attestation_state":"computed","canonical_record":{"metadata":{"abstract_canon_sha256":"2eaa5349d854c7492efc142157bc5c9f48f86f51a87be5a50bf1aacb9c738a5e","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AG","submitted_at":"2015-08-27T20:04:13Z","title_canon_sha256":"077af3832e24434e90365ede52e6da70fafe771bbbf5e72f52afbc16281daa62"},"schema_version":"1.0","source":{"id":"1508.06991","kind":"arxiv","version":4}},"canonical_sha256":"5c97a344f2703604d6f898bae9e103b9ba8a1e88577b96d853c98ccc3ab970ba","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"5c97a344f2703604d6f898bae9e103b9ba8a1e88577b96d853c98ccc3ab970ba","first_computed_at":"2026-05-17T23:59:26.805140Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-17T23:59:26.805140Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"Xz+wz5Bo8mnnUJlO89wNARhpAeZXL2tncDCvfSrO1V8NSD1gNvU/bOYp6afd3iogrTDSeg9I5uIADjpc/cFhBw==","signature_status":"signed_v1","signed_at":"2026-05-17T23:59:26.805637Z","signed_message":"canonical_sha256_bytes"},"source_id":"1508.06991","source_kind":"arxiv","source_version":4}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:52f4ddeb5687d397b2a06ea2c3974252efb9c41c98b03aa7bfe35df01bc29d51","sha256:a9cb7abf2ae8aab7c1c1bfbd9d87135fb65ddef3d1568b65b6b4067edf582c96"],"state_sha256":"1b44f88c25f83d4ac154a3073f41e4b8cdd0532bd4dea68e6630aef067d763f9"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"kg6Mkyu5/k/Ewa6VKuwJeyknvzvNE+0DMQmedtzHLLYV/dcaIEiTqH7V7/WvzRITW5jyN+sVz6hR5hbQTO18Dg==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-04T21:28:07.904253Z","bundle_sha256":"d2faf68a8501628afb24d762777996f6ae21deba159e836d1209758cc62ab1c0"}}