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These are compactifications of $\\mathcal{M}_{g,n}$ and $M_{g,n}$ respectively, obtained by assigning rational weights $A = (a_{1},...,a_{n})$, $0< a_{i} \\leq 1$ to the markings; they are defined over $\\mathbb{Z}$, and therefore over any field. We study the first order infinitesimal deformations of $\\overline{\\mathcal{M}}_{g,A[n]}$ and $\\overline{M}_{g,A[n]}$. In particular, we show that $\\overline{M}_{0,A[n]}$ is rigid over any field, if $g\\ge"},"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":"1701.05861","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AG","submitted_at":"2017-01-20T17:08:36Z","cross_cats_sorted":[],"title_canon_sha256":"7f77af5ca4da424727e79fc4329ba9f661cdf187664c76b1cef2277ae0b4c295","abstract_canon_sha256":"c795f80ac4cd54a914e405407cbe10cb751139c1d8e170227f922aae9e79b475"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T00:52:24.505126Z","signature_b64":"2wuzD+gPKJ0mzrWWPEpEqALdVEP/ccpD4CFqTPZWBKyI+BP59MXJs5Wbb0wYSBmOIWwgZxpf4MThqnrR3CwODQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"937d06d22e1b8dbe95fca8ad5a4879dd4c3dc9d4d1cf9083cc4cc1c8823d1eed","last_reissued_at":"2026-05-18T00:52:24.504465Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T00:52:24.504465Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"On the rigidity of moduli of weighted pointed stable curves","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.AG","authors_text":"Alex Massarenti, Barbara Fantechi","submitted_at":"2017-01-20T17:08:36Z","abstract_excerpt":"Let $\\overline{\\mathcal{M}}_{g,A[n]}$ be the Hassett moduli stack of weighted stable curves, and let $\\overline{M}_{g,A[n]}$ be its coarse moduli space. These are compactifications of $\\mathcal{M}_{g,n}$ and $M_{g,n}$ respectively, obtained by assigning rational weights $A = (a_{1},...,a_{n})$, $0< a_{i} \\leq 1$ to the markings; they are defined over $\\mathbb{Z}$, and therefore over any field. We study the first order infinitesimal deformations of $\\overline{\\mathcal{M}}_{g,A[n]}$ and $\\overline{M}_{g,A[n]}$. In particular, we show that $\\overline{M}_{0,A[n]}$ is rigid over any field, if $g\\ge"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1701.05861","kind":"arxiv","version":1},"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"},"aliases":[{"alias_kind":"arxiv","alias_value":"1701.05861","created_at":"2026-05-18T00:52:24.504572+00:00"},{"alias_kind":"arxiv_version","alias_value":"1701.05861v1","created_at":"2026-05-18T00:52:24.504572+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1701.05861","created_at":"2026-05-18T00:52:24.504572+00:00"},{"alias_kind":"pith_short_12","alias_value":"SN6QNURODOG3","created_at":"2026-05-18T12:31:43.269735+00:00"},{"alias_kind":"pith_short_16","alias_value":"SN6QNURODOG35FP4","created_at":"2026-05-18T12:31:43.269735+00:00"},{"alias_kind":"pith_short_8","alias_value":"SN6QNURO","created_at":"2026-05-18T12:31:43.269735+00:00"}],"events":[],"event_summary":{},"paper_claims":[],"inbound_citations":{"count":0,"internal_anchor_count":0,"sample":[]},"formal_canon":{"evidence_count":0,"sample":[],"anchors":[]},"links":{"html":"https://pith.science/pith/SN6QNURODOG35FP4VCWVUSDZ3V","json":"https://pith.science/pith/SN6QNURODOG35FP4VCWVUSDZ3V.json","graph_json":"https://pith.science/api/pith-number/SN6QNURODOG35FP4VCWVUSDZ3V/graph.json","events_json":"https://pith.science/api/pith-number/SN6QNURODOG35FP4VCWVUSDZ3V/events.json","paper":"https://pith.science/paper/SN6QNURO"},"agent_actions":{"view_html":"https://pith.science/pith/SN6QNURODOG35FP4VCWVUSDZ3V","download_json":"https://pith.science/pith/SN6QNURODOG35FP4VCWVUSDZ3V.json","view_paper":"https://pith.science/paper/SN6QNURO","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1701.05861&json=true","fetch_graph":"https://pith.science/api/pith-number/SN6QNURODOG35FP4VCWVUSDZ3V/graph.json","fetch_events":"https://pith.science/api/pith-number/SN6QNURODOG35FP4VCWVUSDZ3V/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/SN6QNURODOG35FP4VCWVUSDZ3V/action/timestamp_anchor","attest_storage":"https://pith.science/pith/SN6QNURODOG35FP4VCWVUSDZ3V/action/storage_attestation","attest_author":"https://pith.science/pith/SN6QNURODOG35FP4VCWVUSDZ3V/action/author_attestation","sign_citation":"https://pith.science/pith/SN6QNURODOG35FP4VCWVUSDZ3V/action/citation_signature","submit_replication":"https://pith.science/pith/SN6QNURODOG35FP4VCWVUSDZ3V/action/replication_record"}},"created_at":"2026-05-18T00:52:24.504572+00:00","updated_at":"2026-05-18T00:52:24.504572+00:00"}