{"paper":{"title":"Distinguishing $\\Bbbk$-configurations","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.AG"],"primary_cat":"math.AC","authors_text":"Adam Van Tuyl, Federico Galetto, Yong-Su Shin","submitted_at":"2017-05-25T14:25:31Z","abstract_excerpt":"A $\\Bbbk$-configuration is a set of points $\\mathbb{X}$ in $\\mathbb{P}^2$ that satisfies a number of geometric conditions. Associated to a $\\Bbbk$-configuration is a sequence $(d_1,\\ldots,d_s)$ of positive integers, called its type, which encodes many of its homological invariants. We distinguish $\\Bbbk$-configurations by counting the number of lines that contain $d_s$ points of $\\mathbb{X}$. In particular, we show that for all integers $m \\gg 0$, the number of such lines is precisely the value of $\\Delta \\mathbf{H}_{m\\mathbb{X}}(m d_s -1)$. Here, $\\Delta \\mathbf{H}_{m\\mathbb{X}}(-)$ is the fi"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1705.09195","kind":"arxiv","version":2},"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"}