{"paper":{"title":"On the Weak Lefschetz Property for Vector Bundles on $\\mathbb P^2$","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.AC"],"primary_cat":"math.AG","authors_text":"Chris Peterson, Gioia Failla, Zachary Flores","submitted_at":"2018-03-27T21:26:33Z","abstract_excerpt":"Let $R=\\mathbb K[x,y,z]$ be a standard graded polynomial ring where $\\mathbb K$ is an algebraically closed field of characteristic zero. Let $M = \\oplus_j M_j$ be a finite length graded $R$-module. We say that $M$ has the Weak Lefschetz Property if there is a homogeneous element $L$ of degree one in $R$ such that the multiplication map $\\times L : M_j \\rightarrow M_{j+1}$ has maximal rank for every $j$. The main result of this paper is to show that if $\\mathcal E$ is a locally free sheaf of rank 2 on $\\mathbb P^2$ then the first cohomology module of $\\mathcal E$, $H^1_*(\\mathbb P^2, \\mathcal E"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1803.10337","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"}