Explicit examples of resolution of (projective) 3-folds over k?

Question

I'm looking for examples of explicit resolutions of (projective) 3-folds over a field k (char 0), with isolated singularities, or at least with smooth singular locus. I've looked in various books and online, but the examples they present have only been for curves and surfaces, for which resolution of singularities is much less complicated. It would also be nice if the exceptional divisor were sufficiently nice, say, with smooth components that intersect transversally (or if the exceptional divisor were itself smooth).

Are there any well-known/easy examples in dimension 3? Because resolution quickly becomes complicated as dimension increases, I'd imagine that examples become harder to come by, although I'm sure many exist!

In general, I seem to not be very good at finding references/papers relevant to specific things I'm looking for. Search engines typically turn up a lot of irrelevant papers, and thumbing through a bunch of books seems time-inefficient. Search engines have turned up a few papers, but I suspect that I can do better than what I've found thus far.

I realize that adding this second part about looking for references may detract from the specific issue I have right now, but I'd rather get better at finding things than ask here when I can't find something.

Answer 1

What kind of examples are you looking for? I mean there are plenty of examples, just take an explicit singularity and start blowing up. What properties are you looking for? If you are looking for behavior that does not happen in low dimensions, then an interesting family of examples come from what is known as a small resolution, that is, a resolution where the exceptional set is not a divisor.

I believe the simplest example of a small resolution is given by a cone over $\mathbb P^1\times \mathbb P^1$. The resolution is given by blowing up the surface that is the cone over one of the ruling curves on $\mathbb P^1\times \mathbb P^1$. Since the (big) cone is smooth away from the vertex, this blow up will not do anything there and over the vertex it will have an exceptional curve which is isomorphic to $\mathbb P^1$ and really corresponds to points on the curve that gave the blown up surface. This has all the nice properties you asked for: the singular set and the exceptional set are both smooth, it's even an isolated singularity. And, of course, you could have blown up the singular point and get the entire $\mathbb P^1\times \mathbb P^1$ as the exceptional divisor. Some similar and more general examples are computed in this paper.

A similar example can be cooked up from resolving cones over products in general.

Perhaps the next example to consider is when the singular set is larger dimensional, but not simply because it is (say) a product of an isolated singularity with something else.

Yet another way to find many interesting examples is by quotients.

Or you could just try to take your favorite projective variety and project it to a smaller dimensional subspace and then try to resolve the singularity. Although this can get really messy really soon so you have to choose the starting variety carefully. (Dolgachev has a paper on general projection surfaces or something like that with generalizations by Steenbrink and Doherty (two papers).)

Answer 2

Almost the same example, arises from blowing up an ordinary double point on any threefold, since the tangent cone is then a cone over a rank 4 quadric, i.e. $\mathbb{P}^1\times \mathbb{P}^1$. This blowup then introduces a copy of $\mathbb{P}^1\times \mathbb{P}^1$ as exceptional locus. Then one can blow down either ruling and get a small resolution.

The interesting part here is how it illustrates the non uniqueness of the small resolution, as there is no preferred way to choose one of the two rulings. As Moishezon said, "suppose while your back is turned some devil comes and changes which way you blow down." This seems to be the first example of a "flop".

This geometry is already quite interesting in the case of a nodal cubic threefold in $\mathbb{P}^4$. There blowing up the node is almost the same as projecting the cubic from the node into $\mathbb{P}^3$. I.e. the projection is birational and well defined on the blowup, mapping the quartic exceptional locus onto a rank 4 quadric surface in $\mathbb{P}^3$. However those lines in the tangent cone to the cubic at the node that actually lie in the cubic threefold, and which are parametrized by a genus 4 curve in $\mathbb{P}^3$, blow down onto the genus 4 curve.

This reveals the original non small resolution of the nodal cubic as the blow up of $\mathbb{P}^3$ along a canonical genus 4 curve, hence a rational variety. From this viewpoint, since the genus 4 curve meets each line on the unique quadric surface containing it 3 times, it seems Moishezon's criterion implies either family of lines can be blown down on the resulting smooth threefold.

This story is part of the original solution of the Luroth problem for threefolds in the paper of Clemens and Griffiths.

Here is an explicit example: the abel map from the 3-fold symmetric product of a curve of genus 4, onto the theta divisor in the jacobian of that curve. The theta divisor has 2 odp's if the curve is non hyperelliptic, and the exceptional locus then is two copies of P^1.

Answer 3

Another, curve related, example would be the secant variety to a smooth curve. If the embedding is 'sufficiently ample', which would mean that it separates 5 points (it's early and no coffee yet, so '5' should be double checked with a reference) then the secant variety is smooth away from the curve and singular along the curve, so the singular locus is smooth. There is a well know resolution which is discussed in papers of Bertram.
A curve related example with isolated singularities would be the theta divisor of a general genus 4 curve or non-general curves of lower genus. Geometry of Algebraic Curves is a reference for that. Roy Smith is another excellent reference for information of this sort !