4 problems

Reid's conjecture. Under these hypotheses,

Let $A$ be an Abelian variety of dimension $g$, and let $X\subset A$ be a smooth ample hypersurface of dimension $n=g-1$. Assume that $(A,X)$ is a general pair giving a polarizatio…

Existence conjecture. There exists a surface of general type whose canonical map has degree $36$.

Degree-three conjecture. Since it seems very difficult to provide a geometric construction of a fake projective plane, the authors conjecture that