  We discuss the isomorphism problem of projective schemes; given two
projective schemes, can we algorithmically decide whether they are isomorphic?
We give affirmative answers in the case of one-dimensional projective schemes,
the case of smooth irreducible varieties with a big canonical sheaf or a big
anti-canonical sheaf, and the case of K3 surfaces with a finite automorphism
group. As related algorithmic problems, we also discuss decidability of
positivity properties of invertible sheaves, and approximation of the nef cone
and the pseudo-effective cone.
