{"paper":{"title":"On varieties of commuting nilpotent matrices","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.AG","authors_text":"Klemen \\v{S}ivic, Nham V. Ngo","submitted_at":"2013-08-20T21:29:12Z","abstract_excerpt":"Let $N(d,n)$ be the variety of all $d$-tuples of commuting nilpotent $n\\times n$ matrices. It is well-known that $N(d,n)$ is irreducible if $d=2$, if $n\\le 3$ or if $d=3$ and $n=4$. On the other hand $N(3,n)$ is known to be reducible for $n\\ge 13$. We study in this paper the reducibility of $N(d,n)$ for various values of $d$ and $n$. In particular, we prove that $N(d,n)$ is reducible for all $d,n\\ge 4$. In the case $d=3$, we show that it is irreducible for $n\\le 6$."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1308.4438","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"}