[{"@context":"http:\/\/schema.org\/","@type":"BlogPosting","@id":"https:\/\/wiki.edu.vn\/jp\/wiki2\/archives\/6879#BlogPosting","mainEntityOfPage":"https:\/\/wiki.edu.vn\/jp\/wiki2\/archives\/6879","headline":"\u30d2\u30eb\u30d9\u30eb\u30c8\u30b9\u30ad\u30fc\u30e0 – Wikipedia","name":"\u30d2\u30eb\u30d9\u30eb\u30c8\u30b9\u30ad\u30fc\u30e0 – Wikipedia","description":"\u539f\u6587\u3068\u6bd4\u3079\u305f\u7d50\u679c\u3001\u3053\u306e\u8a18\u4e8b\u306b\u306f\u591a\u6570\uff08\u5c11\u306a\u304f\u3068\u30825\u500b\u4ee5\u4e0a\uff09\u306e\u8aa4\u8a33\u304c\u3042\u308b\u3053\u3068\u304c\u5224\u660e\u3057\u3066\u3044\u307e\u3059\u3002\u60c5\u5831\u306e\u5229\u7528\u306b\u306f\u6ce8\u610f\u3057\u3066\u304f\u3060\u3055\u3044\u3002\u6b63\u78ba\u306a\u8868\u73fe\u306b\u6539\u8a33\u3067\u304d\u308b\u65b9\u3092\u6c42\u3081\u3066\u3044\u307e\u3059\u3002 \u4ee3\u6570\u5e7e\u4f55\u5b66\u3067\u306f\u3001\u30d2\u30eb\u30d9\u30eb\u30c8\u30b9\u30ad\u30fc\u30e0\uff08\u82f1: Hilbert scheme\uff09\u3068\u306f\u3001\u5468\u591a\u69d8\u4f53\uff08\u82f1\u8a9e\u7248\uff09(Chow variety)\u3092\u7cbe\u5bc6\u5316\u3057\u305f\u3042\u308b\u5c04\u5f71\u7a7a\u9593\uff08\u3088\u308a\u4e00\u822c\u7684\u306b\u306f\u5c04\u5f71\u30b9\u30ad\u30fc\u30e0\uff09\u306e\u9589\u90e8\u5206\u30b9\u30ad\u30fc\u30e0\u306e\u30d1\u30e9\u30e1\u30fc\u30bf\u7a7a\u9593\u3067\u3042\u308b\u30b9\u30ad\u30fc\u30e0\u3067\u3042\u308b\u3002\u30d2\u30eb\u30d9\u30eb\u30c8\u30b9\u30ad\u30fc\u30e0\u306f\u3001\u30d2\u30eb\u30d9\u30eb\u30c8\u591a\u9805\u5f0f\u306b\u5bfe\u5fdc\u3059\u308b\u9589\u90e8\u5206\u30b9\u30ad\u30fc\u30e0\uff08\u82f1\u8a9e\u7248\uff09(closed subscheme)\u306e\u5171\u901a\u70b9\u3092\u6301\u305f\u306a\u3044\u5408\u4f75\u3067\u3042\u308b\u3002\u30d2\u30eb\u30d9\u30eb\u30c8\u30b9\u30ad\u30fc\u30e0\u306e\u57fa\u672c\u7406\u8ad6\u306f\u3001(Alexander Grothendieck\u00a01961)\u306b\u3088\u308a\u958b\u767a\u3055\u308c\u305f\u3002\u5e83\u4e2d\u306e\u4f8b\uff08\u82f1\u8a9e\u7248\uff09\u306f\u3001\u975e\u5c04\u5f71\u591a\u69d8\u4f53\u306f\u30d2\u30eb\u30d9\u30eb\u30c8\u30b9\u30ad\u30fc\u30e0\u3092\u5fc5\u305a\u3057\u3082\u6301\u305f\u306a\u3044\u3053\u3068\u3092\u793a\u3057\u3066\u3044\u308b\u3002 \u5c04\u5f71\u7a7a\u9593\u306e\u30d2\u30eb\u30d9\u30eb\u30c8\u30b9\u30ad\u30fc\u30e0[\u7de8\u96c6] Pn \u306e\u30d2\u30eb\u30d9\u30eb\u30c8\u30b9\u30ad\u30fc\u30e0 Hilb(n) \u306f\u3001\u6b21\u306e\u610f\u5473\u3067\u5c04\u5f71\u7a7a\u9593\u306e\u9589\u30b9\u30ad\u30fc\u30e0\u3092\u5206\u985e\u3059\u308b\u3002 \u4efb\u610f\u306e\u5c40\u6240\u30cd\u30fc\u30bf\u30fc\u30b9\u30ad\u30fc\u30e0\uff08\u82f1\u8a9e\u7248\uff09(locally Noetherian scheme) S \u306b\u5bfe\u3057\u3001\u30d2\u30eb\u30d9\u30eb\u30c8\u30b9\u30ad\u30fc\u30e0\u306e S \u306b\u5024\u3092\u6301\u3064\u70b9\u306e\u96c6\u5408 Hom(S,","datePublished":"2020-09-28","dateModified":"2020-09-28","author":{"@type":"Person","@id":"https:\/\/wiki.edu.vn\/jp\/wiki2\/archives\/author\/lordneo#Person","name":"lordneo","url":"https:\/\/wiki.edu.vn\/jp\/wiki2\/archives\/author\/lordneo","image":{"@type":"ImageObject","@id":"https:\/\/secure.gravatar.com\/avatar\/c9645c498c9701c88b89b8537773dd7c?s=96&d=mm&r=g","url":"https:\/\/secure.gravatar.com\/avatar\/c9645c498c9701c88b89b8537773dd7c?s=96&d=mm&r=g","height":96,"width":96}},"publisher":{"@type":"Organization","name":"Enzyklop\u00e4die","logo":{"@type":"ImageObject","@id":"https:\/\/wiki.edu.vn\/wiki4\/wp-content\/uploads\/2023\/08\/download.jpg","url":"https:\/\/wiki.edu.vn\/wiki4\/wp-content\/uploads\/2023\/08\/download.jpg","width":600,"height":60}},"image":{"@type":"ImageObject","@id":"https:\/\/upload.wikimedia.org\/wikipedia\/commons\/thumb\/b\/b2\/Blue_question_mark.svg\/30px-Blue_question_mark.svg.png","url":"https:\/\/upload.wikimedia.org\/wikipedia\/commons\/thumb\/b\/b2\/Blue_question_mark.svg\/30px-Blue_question_mark.svg.png","height":"30","width":"30"},"url":"https:\/\/wiki.edu.vn\/jp\/wiki2\/archives\/6879","about":["Wiki"],"wordCount":4694,"articleBody":"\u539f\u6587\u3068\u6bd4\u3079\u305f\u7d50\u679c\u3001\u3053\u306e\u8a18\u4e8b\u306b\u306f\u591a\u6570\uff08\u5c11\u306a\u304f\u3068\u30825\u500b\u4ee5\u4e0a\uff09\u306e\u8aa4\u8a33\u304c\u3042\u308b\u3053\u3068\u304c\u5224\u660e\u3057\u3066\u3044\u307e\u3059\u3002\u60c5\u5831\u306e\u5229\u7528\u306b\u306f\u6ce8\u610f\u3057\u3066\u304f\u3060\u3055\u3044\u3002\u6b63\u78ba\u306a\u8868\u73fe\u306b\u6539\u8a33\u3067\u304d\u308b\u65b9\u3092\u6c42\u3081\u3066\u3044\u307e\u3059\u3002\u4ee3\u6570\u5e7e\u4f55\u5b66\u3067\u306f\u3001\u30d2\u30eb\u30d9\u30eb\u30c8\u30b9\u30ad\u30fc\u30e0\uff08\u82f1: Hilbert scheme\uff09\u3068\u306f\u3001\u5468\u591a\u69d8\u4f53\uff08\u82f1\u8a9e\u7248\uff09(Chow variety)\u3092\u7cbe\u5bc6\u5316\u3057\u305f\u3042\u308b\u5c04\u5f71\u7a7a\u9593\uff08\u3088\u308a\u4e00\u822c\u7684\u306b\u306f\u5c04\u5f71\u30b9\u30ad\u30fc\u30e0\uff09\u306e\u9589\u90e8\u5206\u30b9\u30ad\u30fc\u30e0\u306e\u30d1\u30e9\u30e1\u30fc\u30bf\u7a7a\u9593\u3067\u3042\u308b\u30b9\u30ad\u30fc\u30e0\u3067\u3042\u308b\u3002\u30d2\u30eb\u30d9\u30eb\u30c8\u30b9\u30ad\u30fc\u30e0\u306f\u3001\u30d2\u30eb\u30d9\u30eb\u30c8\u591a\u9805\u5f0f\u306b\u5bfe\u5fdc\u3059\u308b\u9589\u90e8\u5206\u30b9\u30ad\u30fc\u30e0\uff08\u82f1\u8a9e\u7248\uff09(closed subscheme)\u306e\u5171\u901a\u70b9\u3092\u6301\u305f\u306a\u3044\u5408\u4f75\u3067\u3042\u308b\u3002\u30d2\u30eb\u30d9\u30eb\u30c8\u30b9\u30ad\u30fc\u30e0\u306e\u57fa\u672c\u7406\u8ad6\u306f\u3001(Alexander Grothendieck\u00a01961)\u306b\u3088\u308a\u958b\u767a\u3055\u308c\u305f\u3002\u5e83\u4e2d\u306e\u4f8b\uff08\u82f1\u8a9e\u7248\uff09\u306f\u3001\u975e\u5c04\u5f71\u591a\u69d8\u4f53\u306f\u30d2\u30eb\u30d9\u30eb\u30c8\u30b9\u30ad\u30fc\u30e0\u3092\u5fc5\u305a\u3057\u3082\u6301\u305f\u306a\u3044\u3053\u3068\u3092\u793a\u3057\u3066\u3044\u308b\u3002\u5c04\u5f71\u7a7a\u9593\u306e\u30d2\u30eb\u30d9\u30eb\u30c8\u30b9\u30ad\u30fc\u30e0[\u7de8\u96c6]Pn \u306e\u30d2\u30eb\u30d9\u30eb\u30c8\u30b9\u30ad\u30fc\u30e0 Hilb(n) \u306f\u3001\u6b21\u306e\u610f\u5473\u3067\u5c04\u5f71\u7a7a\u9593\u306e\u9589\u30b9\u30ad\u30fc\u30e0\u3092\u5206\u985e\u3059\u308b\u3002\u4efb\u610f\u306e\u5c40\u6240\u30cd\u30fc\u30bf\u30fc\u30b9\u30ad\u30fc\u30e0\uff08\u82f1\u8a9e\u7248\uff09(locally Noetherian scheme) S \u306b\u5bfe\u3057\u3001\u30d2\u30eb\u30d9\u30eb\u30c8\u30b9\u30ad\u30fc\u30e0\u306e S \u306b\u5024\u3092\u6301\u3064\u70b9\u306e\u96c6\u5408Hom(S, Hilb(n))\u306f\u3001S \u4e0a\u306b\u5e73\u5766\uff08\u82f1\u8a9e\u7248\uff09(flat)\u3067\u3042\u308b Pn \u00d7 S \u306e\u9589\u30b9\u30ad\u30fc\u30e0\u306e\u96c6\u5408\u306b\u81ea\u7136\u306b\u540c\u578b\u3068\u306a\u308b\u3002S \u4e0a\u306b\u5e73\u5766\u306a Pn \u00d7 S \u306e\u9589\u90e8\u5206\u30b9\u30ad\u30fc\u30e0\u306f\u3001\u975e\u516c\u5f0f\u306b\u306f\u3001S \u306b\u3088\u308a\u30d1\u30e9\u30e1\u30c8\u30e9\u30a4\u30ba\u3055\u308c\u305f\u5c04\u5f71\u7a7a\u9593\u306e\u90e8\u5206\u30b9\u30ad\u30fc\u30e0\u306e\u65cf\u3068\u8003\u3048\u308b\u3053\u3068\u304c\u3067\u304d\u308b\u3002\u30d2\u30eb\u30d9\u30eb\u30c8\u30b9\u30ad\u30fc\u30e0 Hilb(n) \u306f\u3001\u30d2\u30eb\u30d9\u30eb\u30c8\u591a\u9805\u5f0f P \u3092\u6301\u3064\u5c04\u5f71\u7a7a\u9593\u306e\u90e8\u5206\u30b9\u30ad\u30fc\u30e0\u306e\u663cb\u5f97\u308b\u3068\u591a\u9805\u5f0f\u306b\u5bfe\u5fdc\u3059\u308b\u90e8\u5206\u3067\u3042\u308b Hilb(n, P) \u306e\u5171\u901a\u90e8\u5206\u3092\u6301\u305f\u306a\u3044\u5408\u4f75\u306b\u5206\u89e3\u3059\u308b\u3002\u3053\u308c\u3089\u306e\u90e8\u5206\u306e\u5404\u3005\u306f\u3001Spec(Z) \u4e0a\u3067\u5c04\u5f71\u7684\u3067\u3042\u308b\u3002\u69cb\u6210[\u7de8\u96c6]\u30b0\u30ed\u30bf\u30f3\u30c7\u30a3\u30a8\u30af\u306f\u3001\u30cd\u30fc\u30bf\u30fc\u30b9\u30ad\u30fc\u30e0 S \u4e0a\u306en-\u6b21\u5143\u5c04\u5f71\u7a7a\u9593\u306e\u30d2\u30eb\u30d9\u30eb\u30c8\u30b9\u30ad\u30fc\u30e0 Hilb(n)S \u3092\u3001\u69d8\u3005\u306a\u5224\u5225\u5f0f\u3092 0 \u3068\u3059\u308b\u3053\u3068\u3067\u5b9a\u7fa9\u3055\u308c\u308b\u30b0\u30e9\u30b9\u30de\u30f3\u591a\u69d8\u4f53\uff08\u82f1\u8a9e\u7248\uff09(Grassmannian)\u306e\u90e8\u5206\u30b9\u30ad\u30fc\u30e0\u3068\u3057\u3066\u5b9a\u7fa9\u3057\u305f\u3002\u30d2\u30eb\u30d9\u30eb\u30c8\u30b9\u30ad\u30fc\u30e0\u306e\u57fa\u672c\u7684\u6027\u8cea\u306f\u3001S \u4e0a\u306e\u30b9\u30ad\u30fc\u30e0 T \u306b\u5bfe\u3057\u3001\u30d2\u30eb\u30d9\u30eb\u30c8\u30b9\u30ad\u30fc\u30e0\u306f\u3001T \u4e0a\u306b\u5e73\u5766\u306a Pn \u00d7S T \u306e\u9589\u90e8\u5206\u30b9\u30ad\u30fc\u30e0\u3068\u306a\u308b T-\u306b\u5024\u3092\u6301\u3064\u70b9\u3092\u6301\u3064\u51fd\u624b\u3092\u8868\u73fe\u3059\u308b\u3002X \u304c n-\u6b21\u5143\u5c04\u5f71\u7a7a\u9593\u306e\u90e8\u5206\u30b9\u30ad\u30fc\u30e0\u3067\u3042\u308c\u3070\u3001X \u306f  \u6b21\u6570\u4ed8\u304d\u90e8\u5206\u3067\u3042\u308b IX(m) \u3092\u6301\u3061 n + 1 \u5909\u6570\u306e\u591a\u9805\u5f0f\u74b0 S \u306e\u6b21\u6570\u4ed8\u304d\u30a4\u30c7\u30a2\u30eb IX \u3078\u5bfe\u5fdc\u3059\u308b\u3002X \u306e\u30d2\u30eb\u30d9\u30eb\u30c8\u591a\u9805\u5f0f P \u306b\u306e\u307f\u4f9d\u5b58\u3059\u308b\u5145\u5206\u5927\u304d\u306a m \u306b\u5bfe\u3057\u3001O(m) \u306b\u4fc2\u6570\u3092\u6301\u3064 X \u306e\u5168\u3066\u306e\u9ad8\u6b21\u30b3\u30db\u30e2\u30ed\u30b8\u30fc\u7fa4\u306f\u30010 \u3068\u306a\u308b\u306e\u3067\u3001\u7279\u306b\u3001IX(m) \u306f\u6b21\u5143 Q(m) \u2212 P(m) \u3092\u6301\u3064\u3002\u3053\u3053\u306e Q \u306f\u5c04\u5f71\u7a7a\u9593\u306e\u30d2\u30eb\u30d9\u30eb\u30c8\u591a\u9805\u5f0f\u3067\u3042\u308b\u3002m \u306e\u5024\u3092\u5145\u5206\u5927\u304d\u304f\u3068\u308b\u3002(Q(m) \u2212 P(m))-\u6b21\u5143\u7a7a\u9593 IX(m) \u306f Q(m)-\u6b21\u5143\u7a7a\u9593 S(m) \u306e\u90e8\u5206\u7a7a\u9593\u3067\u3042\u308b\u306e\u3067\u3001\u30b0\u30e9\u30b9\u30de\u30f3\u591a\u69d8\u4f53 Gr(Q(m) \u2212 P(m), Q(m)) \u306e\u70b9\u3092\u8868\u73fe\u3059\u308b\u3002\u3053\u306e\u3053\u3068\u306f\u3001\u30d2\u30eb\u30d9\u30eb\u30c8\u591a\u9805\u5f0f P \u306b\u5bfe\u5fdc\u3059\u308b\u30d2\u30eb\u30d9\u30eb\u30c8\u30b9\u30ad\u30fc\u30e0\u306e\u90e8\u5206\u306e\u30b0\u30e9\u30b9\u30de\u30f3\u591a\u69d8\u4f53\u3078\u306e\u57cb\u3081\u8fbc\u307f\u3092\u4e0e\u3048\u308b\u3002\u3053\u306e\u57cb\u3081\u8fbc\u307f\u306e\u50cf\u306e\u30b9\u30ad\u30fc\u30e0\u306e\u69cb\u9020\u3092\u8a18\u8ff0\u3059\u308b\u3053\u3068\u304c\u6b8b\u3063\u3066\u3044\u308b\u3002\u8a00\u3044\u63db\u3048\u308b\u3068\u3001\u305d\u308c\u306b\u5bfe\u5fdc\u3059\u308b\u30a4\u30c7\u30a2\u30eb\u306e\u5143\u3092\u5145\u5206\u8a18\u8ff0\u3059\u308b\u3053\u3068\u304c\u6b8b\u3063\u3066\u3044\u308b\u3002\u305d\u306e\u3088\u3046\u306a\u5143\u306f\u5199\u50cf IX(m) \u2297 S(k) \u2192 S(k + m) \u304c\u3001\u6b63\u306e k \u306b\u5bfe\u3057\u591a\u304f\u3068\u3082\u30e9\u30f3\u30af dim(IX(k + m)) \u3092\u6301\u3064\u6761\u4ef6\u306b\u3088\u308a\u4e0e\u3048\u3089\u308c\u308b\u3002\u3053\u306e\u6761\u4ef6\u306f\u69d8\u3005\u306a\u5224\u5225\u5f0f\u306e\u6d88\u6ec5\u3068\u540c\u5024\u3067\u3042\u308b\u3002\uff08\u3055\u3089\u306b\u6ce8\u610f\u6df1\u304f\u5206\u6790\u3059\u308b\u3068\u3001k = 1 \u3092\u53d6\u308b\u3060\u3051\u3067\u5341\u5206\u3067\u3042\u308b\u3053\u3068\u304c\u5206\u304b\u308b\u3002\uff09\u5909\u5f62[\u7de8\u96c6]\u30d2\u30eb\u30d9\u30eb\u30c8\u30b9\u30ad\u30fc\u30e0 Hilb(X)S \u306f\u540c\u3058\u65b9\u6cd5\u3067\u4efb\u610f\u306e\u5c04\u5f71\u30b9\u30ad\u30fc\u30e0 X \u306b\u5bfe\u3057\u5b9a\u7fa9\u3055\u308c\u3001\u69cb\u6210\u3055\u308c\u308b\u3002\u975e\u516c\u5f0f\u306b\u306f\u3001\u3053\u306e\u70b9\u306f X \u306e\u9589\u70b9\u306b\u5bfe\u5fdc\u3057\u3066\u3044\u308b\u3002\u6027\u8cea[\u7de8\u96c6]Macaulay (1927) \u3067\u306f\u3001\u591a\u9805\u5f0f\u306e\u30d2\u30eb\u30d9\u30eb\u30c8\u30b9\u30ad\u30fc\u30e0 Hilb(n, P) \u304c\u7a7a\u3067\u306f\u306a\u3044\u3053\u3068\u304c\u793a\u3055\u308c\u3001Hartshorne (1966) \u306f\u3001Hilb(n, P) \u304c\u7a7a\u3067\u306a\u3044\u306a\u3089\u3070\u7dda\u578b\u9023\u7d50\u3067\u3042\u308b\u3053\u3068\u304c\u793a\u3055\u308c\u305f\u3002\u5f93\u3063\u3066\u3001\u5c04\u5f71\u7a7a\u9593\u306e 2\u3064\u306e\u90e8\u5206\u30b9\u30ad\u30fc\u30e0\u304c\u540c\u3058\u30d2\u30eb\u30d9\u30eb\u30c8\u30b9\u30ad\u30fc\u30e0\u306e\u9023\u7d50\u6210\u5206\u3068\u306a\u308b\u3053\u3068\u3068\u3001\u305d\u308c\u3089\u304c\u540c\u3058\u30d2\u30eb\u30d9\u30eb\u30c8\u591a\u9805\u5f0f\u3092\u6301\u3064\u3053\u3068\u3068\u306f\u540c\u5024\u3067\u3042\u308b\u3002\u30d2\u30eb\u30d9\u30eb\u30c8\u30b9\u30ad\u30fc\u30e0\u306f\u3001\u5168\u3066\u306e\u70b9\u3067\u88ab\u7d04\u3067\u306f\u306a\u3044\u65e2\u7d04\u6210\u5206\u306e\u3088\u3046\u306b\u3001\u60aa\u3044\u7279\u7570\u70b9\u3092\u6301\u3064\u3053\u3068\u304c\u3042\u308b\u3002\u305d\u308c\u3089\u306f\u4e88\u671f\u305b\u306c\u9ad8\u6b21\u5143\u306e\u65e2\u7d04\u6210\u5206\u3092\u6301\u3064\u3053\u3068\u3082\u3042\u308b\u3002\u4f8b\u3048\u3070\u3001\u6b21\u5143 n \u306e\u30b9\u30ad\u30fc\u30e0\u306e d \u500b\u306e\u70b9\u306e\u30d2\u30eb\u30d9\u30eb\u30c8\u30b9\u30ad\u30fc\u30e0\uff08\u3082\u3046\u5c11\u3057\u6b63\u78ba\u306b\u306f\u3001\u6b21\u5143 0\u3067\u9577\u3055 d \u306e\u90e8\u5206\u30b9\u30ad\u30fc\u30e0\uff09\u306f\u3001\u6b21\u5143 dn \u3092\u6301\u3064\u3053\u3068\u304c\u671f\u5f85\u3055\u308c\u308b\u304c\u3001n \u2265 3 \u306e\u5834\u5408\u306b\u306f\u65e2\u7d04\u6210\u5206\u304c\u3082\u3063\u3068\u5927\u304d\u306a\u6b21\u5143\u3092\u6301\u3064\u3053\u3068\u304c\u3042\u308a\u3046\u308b\u3002\u591a\u69d8\u4f53\u4e0a\u306e\u70b9\u306e\u30d2\u30eb\u30d9\u30eb\u30c8\u30b9\u30ad\u30fc\u30e0[\u7de8\u96c6]\u30d2\u30eb\u30d9\u30eb\u30c8\u30b9\u30ad\u30fc\u30e0\u306f\u3001\u30b9\u30ad\u30fc\u30e0\u4e0a\u306e 0-\u6b21\u5143\u306e\u90e8\u5206\u30b9\u30ad\u30fc\u30e0\u306e\u7a74\u306e\u3042\u3044\u305f\u30d2\u30eb\u30d9\u30eb\u30c8\u30b9\u30ad\u30fc\u30e0(punctual Hilbert scheme)\u3068\u547c\u3070\u308c\u308b\u3053\u3068\u304c\u3042\u308b\u3002\u975e\u516c\u5f0f\u306b\u306f\u3001\u3053\u306e\u3053\u3068\u306f\u3001\u3044\u304f\u3064\u304b\u306e\u70b9\u304c\u91cd\u306a\u308b\u3068\u304d\u306b\u975e\u5e38\u306b\u9593\u9055\u3063\u305f\u7406\u89e3\u3092\u751f\u307f\u51fa\u3059\u306e\u3067\u3042\u308b\u304c\u3001\u30b9\u30ad\u30fc\u30e0\u306e\u4e0a\u306e\u70b9\u306e\u6709\u9650\u500b\u306e\u96c6\u5408\u306e\u3088\u3046\u306a\u3082\u306e\u3092\u60f3\u5b9a\u3059\u308b\u3053\u3068\u304c\u3067\u304d\u308b\u3002\u4efb\u610f\u306e0-\u6b21\u5143\u30b9\u30ad\u30fc\u30e0\u3092\u95a2\u9023\u3059\u308b 0-\u30b5\u30a4\u30af\u30eb\u3068\u53d6\u308b\u3053\u3068\u306b\u3088\u308a\u3001\u70b9\u306e\u88ab\u7d04\u306a\u30d2\u30eb\u30d9\u30eb\u30c8\u30b9\u30ad\u30fc\u30e0\u304b\u3089\u30b5\u30a4\u30af\u30eb\u306e\u5468\u591a\u69d8\u4f53\u3078\u306e\u30d2\u30eb\u30d9\u30eb\u30c8\u30fb\u5468\u306e\u5c04(Hilbert-Chow morphism)\u304c\u5b58\u5728\u3059\u308b\u3002(Fogarty\u00a01968, 1969, 1973).M \u4e0a\u306e n \u500b\u306e\u70b9\u306e\u30d2\u30eb\u30d9\u30eb\u30c8\u30b9\u30ad\u30fc\u30e0 M[n] \u306f\u3001M \u306e n-\u91cd\u5bfe\u79f0\u7a4d\u3078\u306e\u81ea\u7136\u306a\u5c04\u3092\u6301\u3063\u3066\u3044\u308b\u3002\u3053\u306e\u5c04\u306f\u6700\u5927 2 \u6b21\u5143\u306e M \u306b\u5bfe\u3057\u3066\u53cc\u6709\u7406\u3067\u3042\u308a\u3001\u6700\u5927 3 \u6b21\u5143\u306e M \u306b\u5bfe\u3057\u3066\u3001\u5927\u304d\u306a n \u306b\u5bfe\u3057\u3066\u53cc\u6709\u7406\u3067\u306f\u306a\u3044\u3002\u4e00\u822c\u306b\u3001\u30d2\u30eb\u30d9\u30eb\u30c8\u30b9\u30ad\u30fc\u30e0\u306f\u53ef\u7d04\u3067\u3001\u5bfe\u79f0\u7a4d\u306e\u6b21\u5143\u3088\u308a\u975e\u5e38\u306b\u5927\u304d\u306a\u6b21\u5143\u306e\u8981\u7d20\u3092\u6301\u3063\u3066\u3044\u308b\u3002\u66f2\u7dda C \uff08\u6b21\u5143\u304c 1 \u3067\u3042\u308b\u8907\u7d20\u591a\u69d8\u4f53\uff09\u4e0a\u306e\u70b9\u306e\u30d2\u30eb\u30d9\u30eb\u30c8\u30b9\u30ad\u30fc\u30e0\u306f\u3001C \u306e\u5bfe\u79f0\u3079\u304d\uff08\u82f1\u8a9e\u7248\uff09(symmetric power)\u306b\u540c\u578b\u3067\u3042\u308b\u3002\u66f2\u9762\u4e0a\u306e n \u500b\u306e\u70b9\u306e\u30d2\u30eb\u30d9\u30eb\u30c8\u30b9\u30ad\u30fc\u30e0\u3082\u3001\u6ed1\u3089\u304b\u3067\u3042\u308b (Grothendieck)\u3002n = 2 \u3067\u3042\u308c\u3070\u3001\u5bfe\u89d2\u3092\u30d6\u30ed\u30fc\u30a2\u30c3\u30d7\u3059\u308b\u3053\u3068\u306b\u3088\u308a\u3001\u3064\u307e\u308a\u3001(x, y) \u21a6 (y, x) \u306b\u3088\u308a\u5f15\u304d\u8d77\u3053\u3055\u308c\u305f Z\/2Z \u3067\u5272\u308b\u3053\u3068\u306b\u3088\u308a\u3001M \u00d7 M \u304c\u5f97\u3089\u308c\u308b\u3002\u30de\u30fc\u30af\u30fb\u30cf\u30a4\u30de\u30f3\uff08\u82f1\u8a9e\u7248\uff09(Mark Haiman)\u306b\u3088\u308b\u65b9\u6cd5\u306f\u3001\u3042\u308b\u30de\u30af\u30c9\u30ca\u30eb\u30c9\u591a\u9805\u5f0f\uff08\u82f1\u8a9e\u7248\uff09(Macdonald polynomial)\u306e\u4fc2\u6570\u306e\u6b63\u5024\u6027\u306e\u8a3c\u660e\u306b\u4f7f\u308f\u308c\u305f\u3002\u6b21\u5143\u304c 3 \u4ee5\u4e0a\u306e\u6ed1\u3089\u304b\u306a\u591a\u69d8\u4f53\u306e\u30d2\u30eb\u30d9\u30eb\u30c8\u30b9\u30ad\u30fc\u30e0\u306f\u3001\u901a\u5e38\u306f\u6ed1\u3089\u304b\u3067\u306f\u306a\u3044\u3002\u30d2\u30eb\u30d9\u30eb\u30c8\u30b9\u30ad\u30fc\u30e0\u3068\u8d85\u30b1\u30fc\u30e9\u30fc\u5e7e\u4f55\u5b66[\u7de8\u96c6]M \u3092 c1 = 0 \u3067\u3042\u308b\u8907\u7d20\u30b1\u30fc\u30e9\u30fc\u66f2\u9762\uff08K3\u66f2\u9762\u3001\u3082\u3057\u304f\u306f\u30c8\u30fc\u30e9\u30b9\uff09\u3068\u3059\u308b\u3068\u3001\u5c0f\u5e73\u306e\u66f2\u9762\u306e\u5206\u985e\u306b\u5f93\u3044\u3001M \u306e\u6a19\u6e96\u30d0\u30f3\u30c9\u30eb\u306f\u81ea\u660e\u3067\u3042\u308b\u3002\u3088\u3063\u3066\u3001M \u306f\u6b63\u5247\u306a\u30b7\u30f3\u30d7\u30ec\u30af\u30c6\u30a3\u30c3\u30af\u5f62\u5f0f\u3092\u6301\u3064\u3002\u85e4\u6728\u660e\uff08\u82f1\u8a9e\u7248\uff09(Akira Fujiki)\uff08n = 2 \u306e\u5834\u5408\uff09\u3068\u30a2\u30eb\u30ca\u30a6\u30fb\u30d9\u30eb\u30f4\u30a3\u30eb\uff08\u82f1\u8a9e\u7248\uff09(Arnaud Beauville)\u306b\u3088\u308a\u3001M[n] \u3082\u6b63\u5247\u306a\u30b7\u30f3\u30d7\u30ec\u30af\u30c6\u30a3\u30c3\u30af\u3068\u306a\u308b\u3053\u3068\u304c\u78ba\u8a8d\u3055\u308c\u305f\u3002n = 2\u306b\u5bfe\u3057\u3066\u306f\u3001\u3053\u306e\u3053\u3068\u306f\u3055\u307b\u3069\u306f\u96e3\u3057\u304f\u306a\u3044\u3002\u5b9f\u969b\u3001M[2] \u306f M \u306e\u4e8c\u91cd\u5bfe\u79f0\u7a4d\u306e\u30d6\u30ed\u30fc\u30a2\u30c3\u30d7\u3067\u3042\u308b\u3002Sym2 M \u306e\u7279\u7570\u70b9\u306f\u3001\u5c40\u6240\u7684\u306b C2 \u00d7 C2\/{\u00b11} \u3068\u540c\u578b\u3067\u3042\u308b\u3002C2\/{\u00b11}  \u306e\u30d6\u30ed\u30fc\u30a2\u30c3\u30d7\u306f\u3001T\u2009\u2217P1(C) \u3067\u3042\u308a\u3001\u3053\u306e\u7a7a\u9593\u306f\u30b7\u30f3\u30d7\u30ec\u30af\u30c6\u30a3\u30c3\u30af\u3067\u3042\u308b\u3002\u3053\u306e\u3053\u3068\u306f\u30b7\u30f3\u30d7\u30ec\u30af\u30c6\u30a3\u30c3\u30af\u5f62\u5f0f\u306f\u81ea\u7136\u306b M[n] \u306e\u4f8b\u5916\u56e0\u5b50\u306e\u6ed1\u3089\u304b\u306a\u90e8\u5206\u3078\u62e1\u5f35\u3055\u308c\u308b\u3002M[n] \u306e\u6b8b\u308a\u306e\u90e8\u5206\u306f\u3001\u30cf\u30eb\u30c8\u30fc\u30af\u30b9\u306e\u62e1\u5f35\u5b9a\u7406\u306b\u3088\u308a\u62e1\u5f35\u3055\u308c\u308b\u3002\u6b63\u5247\u306a\u30b7\u30f3\u30d7\u30ec\u30af\u30c6\u30a3\u30c3\u30af\u30b1\u30fc\u30e9\u30fc\u591a\u69d8\u4f53\u306f\u3001\u8d85\u30b1\u30fc\u30e9\u30fc\u3067\u3042\u308b\u3053\u3068\u306f\u3001\u30ab\u30e9\u30d3\u30fb\u30e4\u30a6\u306e\u5b9a\u7406\u3088\u308a\u5f97\u3089\u308c\u308b\u3002K3\u66f2\u9762\u3084 4-\u6b21\u5143\u6b21\u30c8\u30fc\u30e9\u30b9\u4e0a\u306e\u70b9\u306e\u30d2\u30eb\u30d9\u30eb\u30c8\u30b9\u30ad\u30fc\u30e0\u306f\u3001\u8d85\u30b1\u30fc\u30e9\u30fc\u591a\u69d8\u4f53\u306e\u4f8b\u3001K3\u66f2\u9762\u306e\u70b9\u306e\u30d2\u30eb\u30d9\u30eb\u30c8\u30b9\u30ad\u30fc\u30e0\u3068\u4e00\u822c\u5316\u3055\u308c\u305f\u30af\u30f3\u30de\u30fc\u591a\u69d8\u4f53\u3092\u3082\u305f\u3089\u3059\u3002\u95a2\u9023\u9805\u76ee[\u7de8\u96c6]\u53c2\u8003\u6587\u732e[\u7de8\u96c6]Beauville, Arnaud (1983), \u201cVari\u00e9t\u00e9s K\u00e4hleriennes dont la premi\u00e8re classe de Chern est nulle\u201d, Journal of Differential Geometry 18 (4):  755\u2013782, MR730926\u00a0I. Dolgachev (2001), “Hilbert scheme”,  in Hazewinkel, Michiel (ed.), Encyclopaedia of Mathematics, Springer, ISBN\u00a0978-1-55608-010-4\u3002Fantechi, Barbara; G\u00f6ttsche, Lothar; Illusie, Luc; Kleiman, Steven L.; Nitsure, Nitin; Vistoli, Angelo (2005), Fundamental algebraic geometry, Mathematical Surveys and Monographs, 123, Providence, R.I.: American Mathematical Society, ISBN\u00a0978-0-8218-3541-8, MR2222646, https:\/\/books.google.co.jp\/books?id=JhDloxGpOA0C&redir_esc=y&hl=ja\u00a0Fogarty, John (1968), \u201cAlgebraic families on an algebraic surface\u201d, American Journal of Mathematics (The Johns Hopkins University Press) 90 (2):  511\u2013521, doi:10.2307\/2373541, JSTOR\u00a02373541, MR0237496, https:\/\/jstor.org\/stable\/2373541\u00a0Fogarty, John (1969), \u201cTruncated Hilbert functors\u201d, Journal f\u00fcr die reine und angewandte Mathematik 234:  65\u201388, MR0244268, http:\/\/gdz.sub.uni-goettingen.de\/no_cache\/en\/dms\/load\/img\/?IDDOC=252601\u00a0Fogarty, John (1973), \u201cAlgebraic families on an algebraic surface. II. The Picard scheme of the punctual Hilbert scheme\u201d, American Journal of Mathematics (The Johns Hopkins University Press) 95 (3):  660\u2013687, doi:10.2307\/2373734, JSTOR\u00a02373734, MR0335512, https:\/\/jstor.org\/stable\/2373734\u00a0G\u00f6ttsche, Lothar (1994), Hilbert schemes of zero-dimensional subschemes of smooth varieties, Lecture Notes in Mathematics, 1572, Berlin, New York: Springer-Verlag, doi:10.1007\/BFb0073491, ISBN\u00a0978-3-540-57814-7, MR1312161\u00a0Grothendieck, Alexander (1961), Techniques de construction et th\u00e9or\u00e8mes d’existence en g\u00e9om\u00e9trie alg\u00e9brique. IV. Les sch\u00e9mas de Hilbert, S\u00e9minaire Bourbaki 221, http:\/\/www.numdam.org\/item?id=SB_1960-1961__6__249_0\u00a0 Reprinted in Adrien Douady, Roger Godement, Alain Guichardet … (1995), S\u00e9minaire Bourbaki, Vol. 6, Paris: Soci\u00e9t\u00e9 Math\u00e9matique de France, pp.\u00a0249\u2013276, ISBN\u00a02-85629-039-6, MR1611822\u00a0Hartshorne, Robin (1966), \u201cConnectedness of the Hilbert scheme\u201d, Publications Math\u00e9matiques de l’IH\u00c9S (29):  5\u201348, MR0213368, http:\/\/www.numdam.org\/item?id=PMIHES_1966__29__5_0\u00a0Macaulay, F. S. (1927), \u201cSome properties of enumeration in the theory of modular systems\u201d, Proceedings L. M. S. Series 2 26:  531\u2013555, doi:10.1112\/plms\/s2-26.1.531\u00a0Mumford, David, Lectures on Curves on an Algebraic Surface, Annals of Mathematics Studies, 59, Princeton University Press, ISBN\u00a0978-0-691-07993-6\u00a0Nakajima, Hiraku (1999), Lectures on Hilbert schemes of points on surfaces, University Lecture Series, 18, Providence, R.I.: American Mathematical Society, ISBN\u00a0978-0-8218-1956-2, MR1711344\u00a0Nitsure, Nitin (2005), \u201cConstruction of Hilbert and Quot schemes\u201d, Fundamental algebraic geometry, Math. Surveys Monogr., 123, Providence, R.I.: American Mathematical Society, pp.\u00a0105\u2013137, arXiv:math\/0504590, MR2223407\u00a0\u5916\u90e8\u30ea\u30f3\u30af[\u7de8\u96c6]"},{"@context":"http:\/\/schema.org\/","@type":"BreadcrumbList","itemListElement":[{"@type":"ListItem","position":1,"item":{"@id":"https:\/\/wiki.edu.vn\/jp\/wiki2\/#breadcrumbitem","name":"Enzyklop\u00e4die"}},{"@type":"ListItem","position":2,"item":{"@id":"https:\/\/wiki.edu.vn\/jp\/wiki2\/archives\/6879#breadcrumbitem","name":"\u30d2\u30eb\u30d9\u30eb\u30c8\u30b9\u30ad\u30fc\u30e0 – Wikipedia"}}]}]