, they can be glued, which results in a proper variety V. Then V has two smooth rational curves L and M lying over c and d such that
Any 2-dimensional smooth complete variety is projective, so 3 is the smallest possible dimension for such an example.
In a projective variety, a nonzero effective cycle has non-zero degree so cannot be algebraically equivalent to 0.
In Hironaka's example the effective cycle consisting of the two exceptional curves is algebraically equivalent to 0.
Choose C and D so that P has an automorphism σ of order 2 acting freely on P and exchanging C and D, and also exchanging c and d. Then the quotient of V by the action of σ is a smooth 3-dimensional algebraic space with an irreducible curve algebraically equivalent to 0.
The quotient does exist as a scheme if every orbit is contained in an affine open subscheme; the counterexample above shows that this technical condition cannot be dropped.
For quasi-projective varieties, it is obvious that any finite subset is contained in an open affine subvariety.
This property fails for Hironaka's example: a two-points set consisting of a point in each of the exceptional curves is not contained in any open affine subvariety.
For Hironaka's variety V over the complex numbers with an automorphism of order 2 as above, the Hilbert functor HilbV/C of closed subschemes is not representable by a scheme, essentially because the quotient by the group of order 2 does not exist as a scheme (Nitsure 2005, p.112).
In other words, this gives an example of a smooth complete variety whose Hilbert scheme does not exist.
Grothendieck showed that the Hilbert scheme always exists for projective varieties.
Pick a non-trivial Z/2Z torsor B → A; for example in characteristic not 2 one could take A and B to be the affine line minus the origin with the map from B to A given by x → x2.
If V is a complete scheme with a fixed point free action of a group of order 2, then descent data for the map V × B → B are given by a suitable isomorphism from V×C to itself, where C = B×AB = B × Z/2Z.
So if this quotient does not exist as a scheme (as in the example above) then the descent data are ineffective.
If X is a scheme of finite type over a field there is a natural map from divisors to line bundles.
Kleiman found an example of a non-reduced and non-projective X for which this map is not surjective as follows.
Take Hironaka's example of a variety with two rational curves A and B such that A+B is numerically equivalent to 0.