Let $X$ be a smooth and projective variety over a field of characteristic zero. Let $Y$ be a normal variety, with finite quotient singularities (an orbifold!) and let $\pi: Y \to X …

Let $X$ ad $Y$ be (noetherian) schemes and let $\mathcal I \subseteq \mathcal O_Y$ be a sheaf of ideals on $Y$. Let $f \colon X \to Y$ be a morphism of schemes. In general the shea …

The following got no answer on mathstackexchange. I believe it not to be hard, but maybe it is a little specialized? All varieties will be over $\mathbb{C}$ and projective unless …

Hi friends, Can anybody help me with the following question? I start with a projective smooth variety $X$ over a field $k$ of a characteristic 0 and $D$ a normal crossing diviso …

Good morning, I would like to ask the following question concerning the desingularisation, but I'm not familiar at all with these notions. We have the following theorem of Hiron …

Let $X, Y$ be projective algebraic surfaces with isolated singularities. Suppose they are diffeomorphic to each other. Denote by $\phi$ the diffeomorphism from $X$ to $Y$. Then doe …

Let $X$ be a smooth degree $d$ ($d>5$) surface in $\mathbb{P}^3$. We now blow up $X$ at a point and embedd it in $\mathbb{P}^3$. The question is does this resulting surface have on …

The segre cubic primal $X\subset P^4$ is the GIT quotient of 6 points on $P^1$. Let $M_{0,6}$ the DM compactification of the moduli of 6-pointed rational curves. The Segre primal $ …

I will use as a reference Hartshorne Prop. II.7.14, the universal property of blow-up. $\tilde{X}$ is the blow up of $X$ along a sheaf of ideals. $Z\to X$ is the morphism that is t …

Let $X$ be a smooth degree $d$ ($d >5$) surface in $\mathbb{P}^3$. Let $\pi:\tilde{X} \to X$ be a blow-up of $X$ at a point. When is it possible to embed $\tilde{X}$ into $\mathbb{ …

Edit: I can rephrase this question this way: When blowing up every point in the $x$-axis in $\mathbb{C}^2$ by means of an inverse limit of finite blowups, how can anything be 'left …

Let $M$ be a manifold (variety, scheme, your favorite object) and let $N_1,N_2$ be two submanifolds (subvarieties, closed subschemes, ideal sheafes, etc.) such that $N_1 \cap N_2 \ …

Kapranov gave a very nice desciption, over $\mathbb{C}$ of the moduli space of stable pointed rational curves $\overline{M}_{0,n}$ as a series of blow-ups of $P^{n-3}$. Does this, …

Let $X$ be a variety over an algebraically closed field with null characteristic. Let $C$ be a smooth subvariety of $X$ of dimension 1, and let $x$ be a point of $C$. We assume tha …

After a good look for anything in the way of an explicit example, and harassing the algebraic geometers in the department, I still am unable to find an answer to my question, so an …