{"id":293,"date":"2025-05-13T20:33:01","date_gmt":"2025-05-13T18:33:01","guid":{"rendered":"http:\/\/blog.mathyuan.com\/?p=293"},"modified":"2025-05-14T10:55:37","modified_gmt":"2025-05-14T08:55:37","slug":"blow-up-of-affine-variety","status":"publish","type":"post","link":"https:\/\/blog.mathyuan.com\/?p=293","title":{"rendered":"\u4eff\u5c04\u7c07\u7684\u5439\u8d77"},"content":{"rendered":"


\n
\nBlow-up of affine varieties<\/span>\n<\/span>\n

\u6458\u8981<\/span>.<\/span> \u8fd9\u91cc\u6211\u4eec\u603b\u7ed3\u4e00\u4e0bblow-up\u7684\u7ed3\u8bba\u548c\u4f8b\u5b50.
\n<\/div>
\n\u76ee\u5f55<\/span>
\n \u76ee\u5f55<\/a><\/span>
 1.  \u4ec0\u4e48\u662fblow up? \u4ec0\u4e48\u662f\u4f8b\u5916\u9664\u5b50?<\/a><\/span>
 2.  \u5b50\u7c07\u7684\u5439\u8d77<\/a><\/span>
 3.  \u4f8b\u5b50: \u5728\u5355\u4e2a\u51fd\u6570\u5904\u5439\u8d77<\/a><\/span>
 4.  \u66f4\u5f3a\u7684\u5305\u542b<\/a><\/span>
 5.  \u5439\u8d77\u53ea\u53d6\u51b3\u4e8e\u6240\u9009\u7406\u60f3<\/a><\/span>
 6.  \u4eff\u5c04\u5e73\u9762\u5439\u8d77\u539f\u70b9<\/a><\/span>
<\/p>\n

1. \u4ec0\u4e48\u662fblow up? \u4ec0\u4e48\u662f\u4f8b\u5916\u9664\u5b50?<\/a><\/span>\n\n\u9996\u5148, \u6211\u4eec\u4ecb\u7ecd\u7684\u662f\u5bf9affine variety $X$, \u5728\u51fd\u6570$f_1,\\ldots, f_r\\in A(X)$\u5904\u7684\u5439\u8d77. \u5c06\u7ed9\u5b9a\u7684\u8fd9\u4e9b\u51fd\u6570\u5408\u6210\u4e3a$U=X\\backslash$\u4e0a\u7684\u51fd\u6570\\[\\begin{aligned}
\nf:U&\\rightarrow\\mathbb{P}^{r-1}\\\\
\nx&\\mapsto [f_1(x):\\ldots:f_r(x)]
\n\\end{aligned}\\] \u4e8e\u662f\u6211\u4eec\u53ef\u4ee5\u5f97\u5230\u51fd\u6570$f$\u7684\u56fe\u50cf \\[
\n\\Gamma_f\\subset U\\times \\mathbb{P}^{r-1}.
\n\\]\u5c3d\u7ba1$\\Gamma_f\\subset U\\times \\mathbb{P}^{r-1}$\u662f\u95ed\u5b50\u96c6, \u4f46\u5374\u672a\u5fc5\u662f$X\\times \\mathbb{P}^{r-1}$\u7684\u4e2d\u7684\u95ed\u96c6. \u4e8e\u662f\u6211\u4eec\u53ef\u4ee5\u5b9a\u4e49$$\\tilde{X}:=\\overline{\\Gamma_f}\\subset X\\times \\mathbb{P}^{r-1}$$\u4e3a$X$\u5728$f_1,\\cdots f_r$\u5904\u7684\u5439\u8d77. \u90a3\u4e48, \u6295\u5f71\u6620\u5c04$X\\times\\mathbb{P}^{r-1}\\rightarrow X$\u81ea\u7136\u5730\u8bf1\u5bfc\u4e00\u4e2a\u4eceblow up $\\tilde{X}$\u6253\u5230$X$\u7684\u6295\u5f71\u6620\u5c04\\[\\pi:\\tilde{X}\\rightarrow X.\\]\u6ce8\u610f\u5230$\\pi|_{\\Gamma_f}: \\Gamma_f\\rightarrow U\\subset X$\u662f\u540c\u6784, \u4e8e\u662f\u53ef\u4ee5\u5b9a\u4e49\u8be5\u5439\u8d77\u7684\u4f8b\u5916\u9664\u5b50 (exceptional divisor)\u4e3a “$\\tilde{X}$\u4e2d\u4e0d\u6253\u5230$U$\u7684\u90e8\u5206”, \u65e2
\n\\begin{aligned}
\n\\tilde{X}\\backslash \\Gamma_f&=\\tilde{X}\\backslash \\pi^{-1}(D(f_1,\\ldots,f_r)) = \\pi^{-1}(X)\\backslash \\pi^{-1}(D(f_1,\\ldots,f_r))\\\\
\n&= \\pi^{-1}(V(f_1,\\ldots,f_r)).
\n\\end{aligned}<\/p>\n

2. \u5b50\u7c07\u7684\u5439\u8d77<\/a><\/span>\n\n\u5982\u679c$Y\\subset X$\u662f\u4eff\u5c04\u7c07$X$\u7684\u95ed\u5b50\u7c07, \u90a3\u4e48\u53d6\u5b9a$f_1,\\ldots,f_r\\in A(X)$, \u8fd9\u4e9b\u51fd\u6570\u4e5f\u80fd\u770b\u4f5c$Y$\u4e0a\u7684\u51fd\u6570, \u6240\u4ee5\u4e5f\u53ef\u4ee5\u5c06$Y$\u5728$f_1,\\ldots,f_r$\u5904\u5439\u8d77, \u5f97\u5230$\\tilde{Y}$. \u95ee\u9898\u5728\u4e8e: $\\tilde{Y}$\u548c$\\tilde{X}$\u662f\u4ec0\u4e48\u5173\u7cfb?<\/p>\n

\u6ce8\u610f\u5230$Y\\times\\mathbb{P}^{r-1}\\subset X\\times\\mathbb{P}^{r-1}$\u662f\u95ed\u5b50\u96c6, \u800c\u95ed\u5b50\u96c6\u7684\u95ed\u5b50\u96c6\u4f9d\u65e7\u662f\u95ed\u5b50\u96c6, \u6240\u4ee5$f$\u5728$Y\\times\\mathbb{P}^{r-1}$\u4e2d\u7684\u56fe\u50cf\u4e4b\u95ed\u5305, \u4e00\u5b9a\u662f\u5728$X\\times\\mathbb{P}^{r-1}$\u4e2d\u7684\u56fe\u50cf\u4e4b\u95ed\u5305\u7684\u81ea\u5df1. \u8fdb\u4e00\u6b65\u5730, $\\Gamma_f(Y)\\subset Y\\times\\mathbb{P}^{r-1}$\u7684\u95ed\u5305$\\overline{\\Gamma_f}$, \u4e8b\u5b9e\u4e0a\u5c31\u662f$\\overline{\\Gamma_f\\cap Y}$\u5728$\\tilde{X}$\u4e2d\u7684\u95ed\u5305. \u6362\u8a00\u4e4b, $$\\tilde{Y}=\\overline{Y\\cap U}.$$<\/p>\n

\u8fd9\u6837\u4e00\u6765, \u5bf9\u4e8e\u4e00\u4e2a\u4eff\u5c04\u7c07$X$, \u5982\u679c\u5b83\u7684\u4e0d\u53ef\u7ea6\u5206\u89e3\u662f$X=X_1\\cup\\cdots\\cup X_n$, \u90a3\u4e48$\\tilde{X}=\\overline{X_1}\\cup\\cdots\\cup\\overline{X_n}$, \u5c3d\u7ba1\u4e0d\u53ef\u7ea6\u4eff\u5c04\u7c07\u7684blow-up\u672a\u5fc5\u4e5f\u4e0d\u53ef\u7ea6(?), \u4f46\u603b\u6709\u8fd9\u4e2a\u8986\u76d6. \u6240\u4ee5\u6211\u4eec\u603b\u662f\u53ea\u9700\u8981\u8003\u8651blow-up of irreducible varieties.<\/p>\n

3. \u4f8b\u5b50: \u5728\u5355\u4e2a\u51fd\u6570\u5904\u5439\u8d77<\/a><\/span>\n\n\u5982\u679c\u6211\u4eec\u53ea\u9009\u53d6\u4e00\u4e2a\u51fd\u6570$f$, \u5e76\u4e14\u9650\u5b9a$X$\u662f\u4e0d\u53ef\u7ea6\u7684\u4eff\u5c04\u7c07, \u90a3\u4e48$X$\u5728$f$\u5904\u7684\u5439\u8d77$$\\tilde{X}\\subset X\\times \\mathbb{P}^0\\cong X.$$ \u8fd9\u65f6,

  1. \u82e5$f=0$, \u5219$U=\\emptyset$. \u6b64\u65f6$\\tilde{X}=\\emptyset$;<\/li>
  2. \u82e5$f\\neq 0$, \u5219$U\\neq\\emptyset$\u662f\u5f00\u96c6. \u56e0\u4e3a$X$\u662f\u4e0d\u53ef\u7ea6\u7684, \u6240\u4ee5$\\Gamma_f=\\bar{U}=X$. \u8fdb\u800c$\\tilde{X}=X$.<\/li><\/ol>\u4ece\u8fd9\u4e2a\u4f8b\u5b50\u6211\u4eec\u53d1\u73b0, \u4e0d\u65e0\u804a\u7684\u5439\u8d77\u81f3\u5c11\u8981\u5bf9\u4e24\u4e2a\u51fd\u6570\u6765\u505a. <\/p>\n

    4. \u66f4\u5f3a\u7684\u5305\u542b<\/a><\/span>\n\n\u4e8b\u5b9e\u4e0a, $\\tilde{X}$\u4e0d\u4ec5\u5305\u542b\u5728$X\\times\\mathbb{P}^{r-1}$\u4e2d, \u6211\u4eec\u8fd8\u80fd\u627e\u5230\u53e6\u4e00\u4e2a\u5305\u542b$\\tilde{X}$\u7684\u95ed\u96c6.
    \n

    \u547d\u9898 1<\/span>.<\/span> \u8bbe$X$\u662f\u4eff\u5c04\u7c07, $\\tilde{X}$\u662f$X$\u5728$f_1,\\ldots,f_r$\u5904\u7684\u5439\u8d77, \u90a3\u4e48
    \n$$\\tilde{X}\\subset \\{(x,y)\\in X\\times \\mathbb{P}^{r-1}: y_if_j(x)=y_jf_i(x)\\}.$$<\/div><\/p>\n

    5. \u5439\u8d77\u53ea\u53d6\u51b3\u4e8e\u6240\u9009\u7406\u60f3<\/a><\/span>\n\n\u4e8b\u5b9e\u4e0a, \u5982\u679c$f_1,\\ldots,f_r$\u548c$g_1,\\ldots,g_s$\u751f\u6210$A(X)$\u4e2d\u7684\u76f8\u540c\u7406\u60f3$I$, \u90a3\u4e48$X$\u5728$f_1,\\ldots,f_r$\u5904\u7684\u5439\u8d77, \u548c\u5728$g_1,\\ldots,g_s$\u5904\u7684\u5439\u8d77\u662f\u5178\u8303\u540c\u6784\u7684 (\u5b58\u5728\u540c\u6784, \u4f7f\u5f97\u67d0\u4e2a\u5305\u542b$\\pi,\\pi’$\u7684\u56fe\u4ea4\u6362)<\/span>. \u6362\u8a00\u4e4b, blow-up\u53ea\u53d6\u51b3\u4e8e\u6240\u9009\u62e9\u51fd\u6570\u751f\u6210\u7684\u7406\u60f3. \u8fd9\u6837\u4e00\u6765, \u5982\u679c$Y\\subset X$\u662f\u95ed\u5b50\u7c07, \u90a3\u4e48\u53ef\u4ee5\u5728$I(Y)$\u5904\u5439\u8d77$X$, \u79f0\u4e3a\u5728$Y$\u5904\u5439\u8d77$X$. \u8fd9\u6837\u4e00\u6765, \u6211\u4eec\u5c31\u771f\u7684\u53ef\u4ee5\u628a\u4e00\u4e2a\u4eff\u5c04\u7c07\u7684\u4e00\u90e8\u5206 “\u5439\u8d77” \u4e86. <\/p>\n

    6. \u4eff\u5c04\u5e73\u9762\u5439\u8d77\u539f\u70b9<\/a><\/span>\n\n\u6700\u5e38\u7528\u7684\u8fd8\u662f\u4eff\u5c04\u5e73\u9762$\\mathbb{A}^n$\u5439\u8d77\u539f\u70b9, \u5373\u5439\u8d77\u5728coordinate functions $x_1,\\ldots,x_n$\u5904. \u6b64\u65f6\u7684\u597d\u5904\u662f: \u6211\u4eec\u4f1a\u5f97\u5230\u4e00\u4e2a\u5177\u4f53\u7684\u7684\u523b\u753b.
    \n

    \u65ad\u8a00 1<\/span>.<\/span> $$\\tilde{\\mathbb{A}^n}=\\{(x,y)\\in\\mathbb{A}^n\\times\\mathbb{P}^{n-1}: y_ix_j=y_jx_i\\}.$$ \u4e5f\u5373\u547d\u98981\u4e2d\u7684\u7b49\u53f7\u6210\u7acb. \u6b64\u65f6, \u8be5\u5439\u8d77\u7684\u4f8b\u5916\u9664\u5b50$\\pi^{-1}(0)\\subset \\{0\\}\\times\\mathbb{P}^{r-1}\\cong\\mathbb{P}^{r-1}$\u662f\u4e00\u4e2a\u5c04\u5f71\u7c07 (projective).<\/div>
    \n<\/p>\n","protected":false},"excerpt":{"rendered":"

    Blow-up\u7684\u57fa\u672c\u5b9a\u4e49\u548c\u4f8b\u5b50 \u7ee7\u7eed\u9605\u8bfb \u4eff\u5c04\u7c07\u7684\u5439\u8d77<\/span>→<\/span><\/a><\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[9],"tags":[],"class_list":["post-293","post","type-post","status-publish","format-standard","hentry","category-notes"],"_links":{"self":[{"href":"https:\/\/blog.mathyuan.com\/index.php?rest_route=\/wp\/v2\/posts\/293","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/blog.mathyuan.com\/index.php?rest_route=\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/blog.mathyuan.com\/index.php?rest_route=\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/blog.mathyuan.com\/index.php?rest_route=\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/blog.mathyuan.com\/index.php?rest_route=%2Fwp%2Fv2%2Fcomments&post=293"}],"version-history":[{"count":69,"href":"https:\/\/blog.mathyuan.com\/index.php?rest_route=\/wp\/v2\/posts\/293\/revisions"}],"predecessor-version":[{"id":384,"href":"https:\/\/blog.mathyuan.com\/index.php?rest_route=\/wp\/v2\/posts\/293\/revisions\/384"}],"wp:attachment":[{"href":"https:\/\/blog.mathyuan.com\/index.php?rest_route=%2Fwp%2Fv2%2Fmedia&parent=293"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/blog.mathyuan.com\/index.php?rest_route=%2Fwp%2Fv2%2Fcategories&post=293"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/blog.mathyuan.com\/index.php?rest_route=%2Fwp%2Fv2%2Ftags&post=293"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}