{"paper":{"title":"On the Nagata Problem","license":"","headline":"","cross_cats":[],"primary_cat":"math.AG","authors_text":"Ziv Ran","submitted_at":"1998-09-18T04:12:20Z","abstract_excerpt":"Nagata has conjectured that the following statement (N_r) holds for all $r\\geq 10$: (N_r) if $P_1,...P_r \\in {\\mathbb P}^2$ are generic points then any plane curve $C$ satisfies $\\sum_1^r mult_{P_i}(C)\\leq \\sqrt{r} deg(C)$. Nagata proved (N_r) whenever $r$ is a perfect square. Here we prove (N_r) provided $r=k^2+\\alpha,1\\leq\\alpha\\leq2k,k\\geq 3$ and either (i) $\\alpha$ is odd and $\\alpha\\geq \\sqrt{2k}$ or (ii) $\\alpha$ is even and at lest 6, and the fractional part of $\\sqrt{r}$ is at most $2(\\sqrt{2}-1)$."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"math/9809101","kind":"arxiv","version":3},"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"}