All Questions

In Mumford's first paper on Surfaces in char $p$ [1], part 2 Step (II), he wants to show that, given an indecomposable curve of canonical type $D$ on a smooth projective surface $F$ with $p_g(F)=0, ...

When I have a projective surface $X$, for simplicity smooth, I can find a simpler smooth surface on its binational class. In this way we find in a finite number of steps the simplest surface $Y$, i.e. ...

Let $E_i\!: y_i^2 = x_i^3 - x_i$, $i = 1, 2, 3$ be three copies of the supersingular elliptic curve in characteristics $3$. Consider on $E_i$ the following automorphism of order $3$: $$ \sigma(x_i,...

I'm seaching for a proof of the theorem below. Do you know any reference? Consider $f:X\rightarrow Y$ projective birational map between normal varieties and $\mathbb{R}$ cartier divisor $D$ whose ...

In the remark on the bottom of page 5 of this paper, the author states that It is well-known fact that the algebra of a Divisor $D$ with $\kappa (X,D) \leq 1$ is finitely generated over $k$. In ...

I have some issues about understanding the contraction of extremal ray in a concrete situation: Let $\mathcal{E}=\mathcal{O}_{\mathbb{P}^1\times \mathbb{P}^1} \oplus \mathcal{O}_{\mathbb{P}^1\times \...