All Questions

Let $G$ be a reductive group over a field $k$ of characteristic zero with maximal split torus $T$, Borel $B \supset T$ and Weyl group $W$. Set $X:=G/B$ and $C_w:=BwB/B \subset X$ for $w \in W$ the ...

Let $X$ be a (smooth) projective variety (over $\mathbb{C}$), and $\mathcal{L}$ an ample line bundle on $X$. I have heard that then $$ X \cong \mathrm{Proj} \left( \bigoplus_{k \ge 0} H^0(X,\mathcal{...

Let $X$ be a smooth projective curve of genus $g$ over $\mathbb{C}$, $K_{X}$ be a cononical sheaf on $X$ and $\mathcal{E}$ be a locally free sheaf on $X$ s.t. $H^{0}(X,\mathcal{E}^{*})=\operatorname{...