In my post on Resolution of Singularities, I talked about three means of resolving singularities of varieties. One, normalization, which resolves codimension one singularities, has been covered. Now, we’re moving on to the real heavy guns. Hironaka proved that blow-ups resolve ALL singularities. And aside from that, blow-ups are useful to construct new varieties out of old.
Though the geometry is of central importance and, I think, came first historically, we’re going to talk algebra first. Let be a ring and be an ideal. Then we define the graded ring to be the blowup algebra of in . Now, we can express this ring as , that is, we take an additional variable, and then adjoin all the elements of multiplied by it to the ring, and take the grading in . Now, for a finitely generated -algebra, we can write the blowup algebra more explicitly.
Let be an ideal and let be generators of as a -algebra. Then we can define a map by and . Now, we look at the kernel of this map. We have to get an ideal which is homogeneous in the , and so we can define a variety in . This is the blowup of the affine variety determined by along the subvariety determined by . Or, actually, the subscheme determined by , because we made no requirement that be radical. It has the nice property that the projection takes the blowup variety to the original affine variety, and is an isomorphism on the open set of points which aren’t in the preimage of .
Before doing any more description of blowups and their general properties, we’re going to give a much more general construction which reduces to the above in the case of an affine variety.
Now, let be a noetherian scheme and let be a coherent sheaf of ideals on . We define to be the graded sheaf of algebras where . This sheaf of algebras is commutative and quasi-coherent, and locally it is finitely generated by over , which means that generates it over .
So once we have a nice graded sheaf of commutative algebras on a scheme, what can we do with it? Well, perhaps we can generalize our earlier notion of Proj, which took a graded ring, and get a way to take sheaves of graded algebras to schemes. We construct the new functor, Proj, first by doing things locally, then gluing it together. So let be an open affine cover of and a graded sheaf of algebras. Then we denote by , and this comes with a natural morphism . A standard gluing argument (rather similar to the one from normalization) lets us patch these maps together to get .
Now, if you got lost in that nasty local construction, well, it happens, and though it’s nice to know that these things always exist, the situations we will be focusing on will be nicer in general. However, there is an alternate useful characterization of blowups which we should mention, and it doesn’t make existence obvious, but it shows that there’s a nice geometric universal property here. Let be any scheme and let a subscheme. Then the blowup of along , which is is the scheme and morphism characterized by the following two properties:
- The scheme in is a Cartier divisor.
- If is any morphism with a Cartier divisor in , then there is a unique morphism with $latx f=\phi\circ g$.
We call the exceptional divisor of the blowup, and we call the center of the blowup.
Now, there’s yet another way to describe a special case of the blowup: if you’re blowing up an affine variety at a single reduced point (that is, no blowup up at , but you can at .) We’ll just assume that the point we want to look at is the origin, because we can always translate and it keeps those pesky numbers from getting in the way. So we’ll blowup at the origin. Take the coordinates on to be and take coordinates on to be . Then the blowup of at the origin is the zero locus of the polynomials in . For any other affine variety, we just stick it into and then take its inverse image in the map .
So, before when I said that blowing up stretches out a subscheme to a hypersurface, I meant really that it becomes a Cartier divisor. Now, the universal property tells us immediately that it’s useless to blow up at a Cartier divisor, because the universal property is already satisfied! Generally the trick is to blow up a higher codimension subvariety or to blow up a Weil divisor that isn’t Cartier.
So now we’ll work out a couple of blowups explicitly, and see how they work. We start with the nodal cubic, with the thought that blowups are supposed to resolve singularities, so lets see it do the simplest one possible. We’ll use that last characterization of blowups, involving affine space, for this, and so we take to be the homogeneous coordinates on the projective line. So the blowup of at the origin is given by , and we have the equation which defines the nodal cubic. These two define the blowup together. For intuition, think of this roughly as taking the nodal cubic, crossing it with a line, and then taking a diagonal slice, so we want to think that the node has been pulled apart by giving the curve a constantly increasing third coordinate as you go from one end to the other.
To see that this actually happens, look at the set on which . Then we have and in . Substitution yields , and so we get , so the preimage has two components. One is given by , and the other is given by and arbitrary. The latter is just the exceptional divisor, and we toss it out, because though it is the preimage of the origin, we’re only concerned with a few points on that preimage. Now, and together give us a ring isomorphic to , by and , so this component is a copy of , and it maps down to the nodal cubic upon projection. One last remark on this example, this component meets the exceptional divisor at two points, , which are the preimages of the node, so we really did pull it apart.
The part that we cared about above is called the proper transform. It is obtained by taking your subvariety and looking at the open subset of it which contains none of the points in the subvariety being blown up along. Then you look at the inverse image of that open set, and take the closure. So above, the proper transform of the nodal cubic is the affine line, and it intersects the exceptional divisor twice.
Now, we can get places by blowing up nonsingular varieties too. Something we’ll come back to is the topic of nonsingular cubic surfaces, which I mentioned in another post which isn’t part of this series. Take the projective plane and then start blowing up points. Make sure that no three of them lie on a line, though. Do this six times, and make sure that the sixth is chosen so that they don’t all lie on any conic. The surface you get here is isomorphic to a nonsingular cubic surface (we will prove this later, unless the plan changes drastically). Now, it’s a classic fact that there are twentyseven lines on such a surface. Well, here’s the accounting for them:
- 6 are the exceptional divisors of the blowup
- 15 are the lines proper transforms of lines joining pairs of blownup points
- 6 are the proper transforms of conics passing through collections of five points.