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They gave the possibilities of the birational models $W$ of the quotients and the branch divisors $B_0$ induced by involutions $\\sigma$ on the surfaces $S$.\n  In this paper we improve and refine the results of Lee and the second named author. We exclude the case of the Kodaira dimension $\\kappa(W)=1$ when the number $k$ of isolated fixed points of an involution $\\sigma$ on $S$ is nine. The possibilities of branch divisors $B_0$ are reduced for the case $k=9$, and ar","authors_text":"Han Zhang, Yifan Chen, YongJoo Shin","cross_cats":[],"headline":"","license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.AG","submitted_at":"2025-07-02T06:48:46Z","title":"Smooth minimal surfaces of general type with $p_g=0, K^2=7$ and involutions"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2507.01405","kind":"arxiv","version":1},"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":"sha256:364f638b5e5342af7ca7ce376ddf034cba99120edc359d1959194a562ab68bef","target":"record","created_at":"2026-07-05T11:30:48Z","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":"4d53206093700c3fd45c21970d7633bf2b32a6e5d44384a1dce7585ecbc0fc1d","cross_cats_sorted":[],"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.AG","submitted_at":"2025-07-02T06:48:46Z","title_canon_sha256":"8f426262ef24ed8aa5214ccccd64669e8456719afc280cd21ad39bcf78bce5bc"},"schema_version":"1.0","source":{"id":"2507.01405","kind":"arxiv","version":1}},"canonical_sha256":"1a944d2179ff120579f153c85baa2e84ca8798279b18063033b1ee5e22340de3","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"1a944d2179ff120579f153c85baa2e84ca8798279b18063033b1ee5e22340de3","first_computed_at":"2026-07-05T11:30:48.259025Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-07-05T11:30:48.259025Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"mI7CBtACtpzO1IZAvybgmnfz7bBRNscKmFcOUW7tln7pSas7pzfiy3maQQcHe8Y1rirZjHGfPLo8LCdtLArkCw==","signature_status":"signed_v1","signed_at":"2026-07-05T11:30:48.259525Z","signed_message":"canonical_sha256_bytes"},"source_id":"2507.01405","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:364f638b5e5342af7ca7ce376ddf034cba99120edc359d1959194a562ab68bef","sha256:b2b9b46193b5c948c47d8915390947718553a0775c27003a6ebce63d090f9136"],"state_sha256":"f076699b58017b291382fea4dccac5902f878d7110e9cf72c538c75aa690dbf8"}