{"paper":{"title":"Deformation Theory and Torus-Fixed Geometry of the Nested Hilbert Scheme of Points","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.AC","math.CO"],"primary_cat":"math.AG","authors_text":"Chenyang Zhao","submitted_at":"2026-06-06T11:50:02Z","abstract_excerpt":"In this thesis, we study the nested Hilbert scheme ((\\mathbb{A}^2)^{[n,n+1]}=\\mathrm{Hilb}^{n,n+1}(\\mathbb{A}^2)) from a combination of deformation theory, torus actions, and Young diagram combinatorics. We first recall the scheme theory and functor basics needed to define Hilbert schemes. We then use a classic result on first-order deformations to identify (T_I(\\mathbb{A}^2)^{[n]}\\cong \\mathrm{Hom}*{\\mathbb{C}[x,y]}(I,\\mathbb{C}[x,y]/I)). For a nested pair (I\\subset J), with (\\dim*{\\mathbb{C}}\\mathbb{C}[x,y]/I=n+1) and (\\dim_{\\mathbb{C}}\\mathbb{C}[x,y]/J=n), the tangent space becomes a compat"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2606.08120","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":""},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2606.08120/integrity.json","findings":[],"available":true,"detectors_run":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938"},"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"}