{"id":"W2153391576","doi":"10.1016/j.aim.2009.10.003","title":"Smooth and irreducible multigraded Hilbert schemes","year":2009,"lang":"en","type":"article","venue":"Advances in Mathematics","topic":"Commutative Algebra and Its Applications","field":"Mathematics","cited_by":21,"is_retracted":false,"has_abstract":false,"ca_institutions":"Queen's University","funders":"Natural Sciences and Engineering Research Council of Canada; University of Warwick; National Science Foundation","keywords":"Hilbert scheme; Mathematics; Hilbert series and Hilbert polynomial; Abelian group; Hilbert–Poincaré series; Conjecture; Polynomial ring; Pure mathematics; Polynomial; Homogeneous; Scheme (mathematics); Ring (chemistry); Hilbert's fourteenth problem; Combinatorics; Hilbert space; Rigged Hilbert space; Mathematical analysis; Reproducing kernel Hilbert space","routes":{"ca_aff":true,"ca_fund":true,"ca_venue":false,"about_ca":false,"invisible_to_affiliation_only":false},"retraction":null,"screen":null,"direct_labels":[],"prediction":{"model_version":"codex-gemma-dda1882f352a","candidate_categories":[],"consensus_categories":[],"category_scores_codex":[0.0002026106,0.0001716617,0.0002829048,0.00009865206,0.00008125309,0.00003084851,0.0001911187,0.00005227156,0.00004689924],"category_scores_gemma":[0.0003831629,0.0001500088,0.00003786117,0.0002709127,0.00008682335,0.000307505,0.00004159816,0.0001512932,0.00001486084],"about_ca_system_candidate":false,"about_ca_system_consensus":false,"about_ca_system_score_codex":0.00002636627,"about_ca_system_score_gemma":0.00001417996,"about_ca_topic_candidate":false,"about_ca_topic_consensus":false,"about_ca_topic_score_codex":0.000001458067,"about_ca_topic_score_gemma":0.00001641649,"domain_scores_codex":[0.9990048,0.00002291092,0.0003669004,0.000223548,0.0001562727,0.0002255433],"domain_scores_gemma":[0.9987611,0.000600212,0.000133274,0.0004058,0.00004025191,0.00005933195],"domain_codex":null,"domain_gemma":null,"domain_candidate":null,"domain_consensus":null,"study_design_codex":"theoretical_or_conceptual","study_design_gemma":"theoretical_or_conceptual","study_design_scores_codex":[0.000004626859,0.0004154352,0.0003103734,0.0001351337,0.000007096375,0.000002421624,0.002451312,0.000004448288,0.0007657424,0.952829,0.0003363962,0.04273802],"study_design_scores_gemma":[0.0003779561,0.00004157794,0.0008987847,0.000109915,0.00001547039,0.000009095108,0.0005323206,0.001147799,0.001301795,0.9815576,0.01381369,0.0001939653],"study_design_candidate":"theoretical_or_conceptual","study_design_consensus":"theoretical_or_conceptual","genre_codex":"empirical","genre_gemma":"empirical","genre_scores_codex":[0.8830708,0.00883573,0.0338892,0.003759584,0.0001164619,0.001465728,0.00001520396,0.0003755576,0.06847177],"genre_scores_gemma":[0.73794,0.0008503494,0.2605037,0.0001198177,0.00003102243,0.00004496417,0.00000285607,0.0000207969,0.0004865478],"genre_candidate":"empirical","genre_consensus":"empirical","teacher_disagreement_score":0.2266145,"threshold_uncertainty_score":0.6117182,"prediction_status":"machine_predicted_unvalidated"},"machine_scores":{"provisional":true,"baseline":true,"maturity_gate_passed":false,"score_opus":0.03230269960109491,"score_gpt":0.3531584922400731,"score_spread":0.3208557926389782,"validation_status":"score_only:v0-immature-baseline","note":"Baseline scores from an immature model (maturity gate not passed). Scores rank; they never assert a category."}}