Blow up along codimension one closed subscheme - MathOverflow

Blow up along codimension one closed subscheme

2010-01-06T20:32:09Z

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.

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.

Answer by Ilya Nikokoshev for Blow up along codimension one closed subscheme

2010-01-06T20:34:57Z

How do you define blow-up? It should be straightforward to show that an explicit construction has relative dimension 0 over $X$.

Update: 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.

Answer by Olivier Benoist for Blow up along codimension one closed subscheme

2010-01-06T21:46:22Z

I think it's not true :

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 :

$$X'=Proj_{A}A[t,u]/(zt-(x+y)u,(x-y)t-zu),$$ [EDIT : THE FORMULA HAS BEEN CORRECTED]

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 !