{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2015:AEH2PAE6NFIEWXPIX4GW4KWCGR","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":"d6dc7064a8dbb096266fd66d62dc4dc46b3950b6012fb16939494742d71f6443","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AG","submitted_at":"2015-12-12T16:43:17Z","title_canon_sha256":"e1041156724af7a0e0c383cb34e5a329f1d2603856392e4b173a59aa498eb3d7"},"schema_version":"1.0","source":{"id":"1512.03933","kind":"arxiv","version":2}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1512.03933","created_at":"2026-05-18T01:12:55Z"},{"alias_kind":"arxiv_version","alias_value":"1512.03933v2","created_at":"2026-05-18T01:12:55Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1512.03933","created_at":"2026-05-18T01:12:55Z"},{"alias_kind":"pith_short_12","alias_value":"AEH2PAE6NFIE","created_at":"2026-05-18T12:29:10Z"},{"alias_kind":"pith_short_16","alias_value":"AEH2PAE6NFIEWXPI","created_at":"2026-05-18T12:29:10Z"},{"alias_kind":"pith_short_8","alias_value":"AEH2PAE6","created_at":"2026-05-18T12:29:10Z"}],"graph_snapshots":[{"event_id":"sha256:150bdb5e8e1a73fbbc8439b49d77e6f350213e106005396dc3bec06cd5a909e4","target":"graph","created_at":"2026-05-18T01:12:55Z","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":"In this paper, we investigate the general notion of the slope for families of curves $f: X \\to Y$. The main result is an answer to the above question when $\\dim Y = 2$, and we prove a lower bound for this new slope in this case over fields of any characteristic. Both the notion and the slope inequality are compatible with the theory for $\\dim Y = 0, 1$ in a very natural way, and this gives a strong evidence that the slope for an $n$-fold fibration of curves $f: X \\to Y$ may be $K_{X/Y}^n / \\mathrm{ch}_{n-1}(f_* \\omega_{X/Y})$.\n  Rather than the usual stability methods, the whole proof of the s","authors_text":"Tong Zhang","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AG","submitted_at":"2015-12-12T16:43:17Z","title":"Slope inequality for families of curves over surfaces"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1512.03933","kind":"arxiv","version":2},"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":"sha256:13c3b2db264b1eaba057191901739b185a78aeab13ddfc6d52102ed201066a52","target":"record","created_at":"2026-05-18T01:12:55Z","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":"d6dc7064a8dbb096266fd66d62dc4dc46b3950b6012fb16939494742d71f6443","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AG","submitted_at":"2015-12-12T16:43:17Z","title_canon_sha256":"e1041156724af7a0e0c383cb34e5a329f1d2603856392e4b173a59aa498eb3d7"},"schema_version":"1.0","source":{"id":"1512.03933","kind":"arxiv","version":2}},"canonical_sha256":"010fa7809e69504b5de8bf0d6e2ac2344d6f4b0576c3a19deed9c34388d953ff","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"010fa7809e69504b5de8bf0d6e2ac2344d6f4b0576c3a19deed9c34388d953ff","first_computed_at":"2026-05-18T01:12:55.106939Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T01:12:55.106939Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"RVl1zQvQ3OEWe/qhX5JOiwoLSTgJWbEeJowZ6XHnDWzeCsDLfhT+ApoIOl/48zjevW4TtKdfuz1Gq51XqNtoCA==","signature_status":"signed_v1","signed_at":"2026-05-18T01:12:55.107272Z","signed_message":"canonical_sha256_bytes"},"source_id":"1512.03933","source_kind":"arxiv","source_version":2}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:13c3b2db264b1eaba057191901739b185a78aeab13ddfc6d52102ed201066a52","sha256:150bdb5e8e1a73fbbc8439b49d77e6f350213e106005396dc3bec06cd5a909e4"],"state_sha256":"126175717c3a1be8c598e8fb306ab3c90dc66413a27cf8003f497ff2a111097c"}