Blow up along codimension one closed subscheme - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-19T23:18:51Z http://mathoverflow.net/feeds/question/10960 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/10960/blow-up-along-codimension-one-closed-subscheme Blow up along codimension one closed subscheme Taisong Jing 2010-01-06T20:32:09Z 2010-01-07T21:57:40Z <p>Suppose X is dimension two locally Noetherian scheme. Y is a closed subscheme of X, with codimension 1. Denote X' to be the blow up of X along Y. Prove that the structure morphism f:X'-->X is a finite morphism. </p> <p>It suffices to show it's quasi-finite according to Zariski's main theorem. But I can't exclude the possibility that an irreducible component of \$f^{-1}(Y)\$ maps to a closed point of Y.</p> http://mathoverflow.net/questions/10960/blow-up-along-codimension-one-closed-subscheme/10961#10961 Answer by Ilya Nikokoshev for Blow up along codimension one closed subscheme Ilya Nikokoshev 2010-01-06T20:34:57Z 2010-01-06T22:14:28Z <p>How do you define blow-up? It should be straightforward to show that an explicit construction has relative dimension 0 over \$X\$. </p> <p><strong>Update:</strong> In the comments I suggest to take the formal two-dimensional ring \$k[[x_1, x_2]]\$ and work with it. This assumes that \$X\$ is smooth. If it's not, then Olivier gave an example of quadratic cone where blowup has relative dimension 1.</p> http://mathoverflow.net/questions/10960/blow-up-along-codimension-one-closed-subscheme/10970#10970 Answer by Olivier Benoist for Blow up along codimension one closed subscheme Olivier Benoist 2010-01-06T21:46:22Z 2010-01-07T21:57:40Z <p>I think it's not true : </p> <p>Let \$X=Spec(A)\$ with \$A=k[x,y,z]/(x^2-y^2-z^2)\$ be a quadratic cone. Let \$Y\$ be a line through the origin of the cone : its ideal is \$I=(z,x-y)\$. We calculate : </p> <p>\$\$X'=Proj_{A}A[t,u]/(zt-(x+y)u,(x-y)t-zu),\$\$ [<strong>EDIT : THE FORMULA HAS BEEN CORRECTED</strong>]</p> <p>where, in the graded \$A\$-algebra \$A+I+I^2+....\$ we denoted \$t\$ and \$u\$ the degree one generators corresponding to \$z\$ and \$x-y\$. Now, quotienting by \$x\$, \$y\$, and \$z\$, we calculate the fiber over the origin of this blow-up It is Proj(k[t,u]), which is a positive-dimensional projective line !</p>