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Let $X=Spec(R)$. Blowing-up $Z=V(I)$ is the same as to look at $Proj$ of the graded ring $R[It]=\oplus_{j\geqslant 0} I^jt^j\subset R[t]$, the Rees ring associated to $I$. Assume $R$ is a domain, inte …

The Hirzebruch surfaces are the $\mathbb{P}^1$ bundles $\mathbb{F_n}$ ($n\geqslant 0$) which can be obtained projectivizing the rank $2$ vector bundles $\mathcal{O}_{\mathbb{P}^1}\oplus \mathcal{O}_{\ …

Let $(k,|\cdot|)$ be an algebraically closed field, complete wrt a (multiplicative) norm as in the framework of the Berkovich's analytic geometry. Given a commutative Banach $k$-algebra $\mathcal{A}\ …

I decided to turn my comment into an answer not because it is complete but because I think it can be of use. Let $z=(z_0:z_1:z_2)$ and $u=(u_1:\ldots:u_k)$ be homogeneous coordinates of $\mathbb{P}^2$ …

Without any stronger hypothesis connecting $f$ and $g$ you should not expect that to happen. The simplest instance is when $(X,o)$ is alrealdy non singular, say dimension 2: $(\mathbb{C}^2,o)$. In thi …

I asked this question a few days ago in MathExchange and received no satisfatory answer. I hope it is well suited for MathOverflow. Suppose I have two linear operators $X,\,Y$ on $\mathbb{C}^n$. Now l …