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An open subset $U$ of a projective surface $Z$ is big if $\mathrm{codim}_Z(Z\setminus U)\geq2$. Let $X$ and $Y$ be smooth complex projective surface. If there exists a birational map $f:X\...

Let $E_i\!: y_i^2 = x_i^3 + a_4x_i + a_6$ be two copies ($i = 1$, $2$) of a supersingular elliptic curve over a finite field $\mathbb{F}_{p^2}$, for odd prime $p > 3$. Consider the Kummer surface $...

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,...

Let $k$ be an algebraically closed field of characteristic $p>0$. I have been trying to find out unsuccessfully if there is a mmp for algebraic surfaces over $k$. I know minimal surfaces are ...

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 \...