{"paper":{"title":"A note on normal triple covers over $\\mathbb{P}^2$ with branch divisors of degree 6","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.AG","authors_text":"Taketo Shirane","submitted_at":"2012-11-12T07:42:27Z","abstract_excerpt":"Let $S$ and $T$ be reduced divisors on $\\mathbb{P}^2$ which have no common components, and $\\Delta=S+2\\,T.$ We assume $\\deg\\Delta=6.$ Let $\\pi:X\\to\\mathbb{P}^2$ be a normal triple cover with branch divisor $\\Delta,$ i.e. $\\pi$ is ramified along $S$ (resp. $T$) with the index 2 (resp. 3). In this note, we show that $X$ is either a $\\mathbb{P}^1$-bundle over an elliptic curve or a normal cubic surface in $\\mathbb{P}^3.$ Consequently, we give a necessary and sufficient condition for $\\Delta$ to be the branch divisor of a normal triple cover over $\\mathbb{P}^2.$"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1211.2526","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"}