All Questions

We know that $H^2(\mathcal{O}_{\mathbb{P}^3})=0$. I am looking for blow-ups $$\pi:X \to \mathbb{P}^3$$ such that $X$ is non-singular and $H^2(\mathcal{O}_X)>0$. Of course, if we blow-up along ...

Let $C$ be a curve and $L$ be a line bundle with sufficiently large degree. Let $C_p$ denote the $p$-th symmetric product of $C$, which consists of all the effective divisors of degree $p$ on $C$. Let ...

I'm studying the paper "Varieties with small Dual Varieties" by L.Ein and in the construction he gives about the $10-$dimensional spinor variety $S_4 \subset \mathbb{P}^{15}$ I'm finding ...

This is basically the last step of problem 5.3.7 in Huybrechts' Complex Geometry. Let $X$ be a complex manifold, $x \in X$, $E$ a holomorphic vector bundle on $X$ and $L$ a positive line bundle. ...

Let $S\subset\mathbb{P}^n$ be a finite set of $s$ reduced points. Let $\mathcal{I}$ be the ideal sheaf of $S$ in $\mathbb{P}^n$. We consider the sheaf $$\mathcal{F}_k:=\mathcal{O}_{\mathbb{P}^n}(kd)\...

Let $X$ be a smooth scheme and $Z\subset X$ a smooth subscheme. Consider the blow-up $$\pi:\widetilde{X}:=Bl_{Z}X\rightarrow X$$ of $X$ along $Z$. What is the relation between the cohomology of the ...