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I'm interested in how do people that work in birational geometry view their field — specifically, what are the kinds of geometric questions (as opposed to commutative-algebraic questions) that ...

Consider the Fermat sextic curve $F: x^6 + y^6 + 1 = 0$ over an algebraically closed field of characteristic $0$. It has the two order $3$ automorphisms $\omega_x(x,y) := (\omega x, y)$ and $\omega_y(...

I am interested in the existence of a surface $X$ over $\mathbb{C}$ with the following properties (or a reason for why one cannot exist): $X$ is slc (and not-normal) There is rational curve $C \...

Assume we work over $\mathbb{C}$. Let $S\subset \mathbb{P}^3$ be a quartic surfaces with 16 nodes (ordinary double points). Then there is a simple principally polarized abelian surface $(A,\theta)$ ...