“I think a blessing/curse of our professions is that the lab is in our mind.” I don’t know who said that, but he’s damn right…
Today I want to talk about a wrong belief I held for the last 2 years. I thought that a variety X over k is projective if and only if each of it’s irreducible components is projective until very recently. I held the wrong belief because the following exercise in Hartshorne’s famous <Algebraic Geometry>: Let X be a scheme over k, a line bundle on X is ample if and only if the restriction on each irreducible components is ample. An problem would be that even on each irreducible component we have an ample line bundle we can not always patch them together (although we can always patch in 1-dimensional case). I came to the following (counter)example in my research.
Let’s take two pieces of P^2 blow up at one point, call them X and Y. There is a ordinary(a usual line not passing thru the point we blown up) line and an exceptional divisor on this blow up of P^2. Let’s identify the exceptional divisor on X with an ordinary line on Y and identify the exceptional divisor on Y with an ordinary line on X. Call the result variety Z, which has two projective irreducible components. One can easily see that any two ample line bundles on X and Y can never be patched together (leave as an exercise to reader(s?)). Does anyone besides me had the same wrong belief before reading this post (Does anyone actually read this post?)? Or is it just me?
This example isn’t coming from nowhere, it is actually a reduction of some special non-archimedean Hopf surface. After a tedious computation one can show some reductions of some more general non-archimedean Hopf surfaces, whose irreducible components are blow up of Hirzebruch surfaces (toric surfaces), are also non-projective.