{"paper":{"title":"The ring of evenly weighted points on the line","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.AC","math.CO"],"primary_cat":"math.AG","authors_text":"Benjamin Howard, Milena Hering","submitted_at":"2012-11-16T16:20:15Z","abstract_excerpt":"Let $M_w = (\\Pj^1)^n \\q \\mathrm{SL}_2$ denote the geometric invariant theory quotient of $(\\Pj^1)^n$ by the diagonal action of $\\mathrm{SL}_2$ using the line bundle $\\mathcal{O}(w_1,w_2,...,w_n)$ on $(\\Pj^1)^n$. Let $R_w$ be the coordinate ring of $M_w$. We give a closed formula for the Hilbert function of $R_w$, which allows us to compute the degree of $M_w$. The graded parts of $R_w$ are certain Kostka numbers, so this Hilbert function computes stretched Kostka numbers. If all the weights $w_i$ are even, we find a presentation of $R_w$ so that the ideal $I$ of this presentation has a quadrat"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1211.3941","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"}