Blow-up and regular sequence I'd like how to deduce, if it's possible that:
the blowup of an affine variety $X$ along $V(g_1,\ldots,g_k)$ is $V(t_i g_j-t_j g_i)_{i,j}\hookrightarrow X\times\text{Proj}(k[t_1,\ldots,t_k])$ if the sequence $(g_1,\ldots,g_k)$ is regular
from the fact that:
if $(g_1,\ldots,g_k)$ is regular genrating an ideal $I$ then
$$ (A/I)[t_1,\ldots,t_k]\simeq\text{gr}_I A=\oplus I^n/I^{n+1} $$
by evaluation on the $g_i$.
For me blowup along $V(I)$ is $\text{Proj}(\oplus I^n)$. As we have $A[t_i]\to \oplus I^n$ surjective by evaluation on the $g_i$ so we have $\text{Proj}(\oplus I^n)\hookrightarrow X\times \mathbb{P}^{k-1}$ closed immersion of equations $\ker(A[t_i]\to\oplus I^n)$ which, by the result, is $I[t_i]$. Here I have problem because I don't see why $I[t_i]$ should be generated by the $g_jt_i-g_it_j$.
 A: The book of Swanson and Huneke is indeed very good for studying defining equations of Rees algebras. It is in section 5.5 page 110.
The central idea is that for some ideals $I=(x_1,\ldots,x_n)$ called of linear type if the defining equations are only given by the relations (the syzygies) between the generator $x_i$ (if $\sum_i a_i x_i$ is a relation then $\sum_i a_i t_i$ is clearly a (linear) equation of the Rees algebra).
It can be proven ideal generated by regular sequence are of linear type. But for regular sequence the Koszul complex is exact so that the relations between the $x_i$ are only the Koszul relations $x_i x_j- x_j x_i=0$ so that the equations of the Rees algebra are the $x_iT_j-x_j T_i=0$.
The key point if so to prove that ideal generated by regular sequence are of linear type. It is more generaly the case if $x_1,\ldots,x_n$ is a d-sequence:
$$ (x_0,\ldots,x_i):x_{i+1}x_j=(x_0,\ldots,x_i):x_j $$
for all $0\leqslant i\leqslant n-1$ and for all $j\geqslant i+1$
It is not difficult to see that regular sequence are d-sequences because $x_{i+1}$ is a regular element in $A/(x_1,\ldots,x_i)$
The fact that d-sequences are of linear type is a direct corolary of Theorem 5.5.4 page 111 in Swanson and Huneke with an half page for proof.
I don't know I there is some more direct proof for the specific case a regular sequence.
