This came up in my research recently.
In geometry, “simple nodes” are the simplest, nicest kinds of curve singularities — they have length 1, they’re resolved after a single blowup, they are the only singularities that occur “generically”, and their scheme-theoretic properties seem generally well-understood and well-behaved. (For example, the moduli space of marked points on stable curves of genus exists because of, among other things, the restriction to only nodal curves.)
Intuitively, to obtain a nodal curve , one takes two points and (on smooth curves, say) and glues the curves together at those two points. This is a minimal gluing — the pushout of the diagram showing the inclusion of a point at each :
In particular, the glued curve has a universal property: if and are two morphisms, such that (considering only the closed points), then and glue to give a unique morphism .
But does the same construction work for, say, gluing a complicated singularity of to another complicated singularity on with ‘minimal contact’?
Turns out, the answer is yes, and in fact more is true:
Proposition. Let be integral schemes over an algebraically closed field , and pick and . Then there exists a scheme , the gluing of to at the points , with the following properties:
- It has two irreducible components, isomorphic to and , which intersect scheme-theoretically in one reduced point;
- It satisfies the universal property stated above — namely, a map is the same as a pair of maps that agree set-theoretically at and .
Moreover, the first property implies the second, and so uniquely characterizes the gluing (up to unique isomorphism).
Corollary. Let be any morphism whose restrictions to and to are isomorphisms. Then is an isomorphism.
(This follows by using the universal property to construct the inverse map back to .)
The first property ends up not being so hard to prove: we look inside , and take the union of the fibers and . This obviously satisfies the first bullet point. (If this is hard to picture when and are highly singular, consider the case where they are projective, and consider them inside . Clearly, the “horizontal fiber” intersects the “vertical fiber” in one reduced point.)
To show that the first property implies the second, here’s what’s going on algebraically. We have a reduced local ring with exactly two minimal prime ideals . (In particular .) The fact that the irreducible components meet scheme-theoretically in one point translates to .
The statement we want is the following: let and be two maps such that , where is the projection map or .
Then there exists a unique homomorphism such that the following diagram commutes:
Constructing this map basically comes down to the standard but slightly-tricky-to-prove short exact sequence
where the first map is restriction and the second is subtraction. In our case, this sequence becomes
In particular, let with images and . Since have the same image in the residue field, by the sequence above, they lift to a unique element . Then the assignment is the desired ring homomorphism.
(There are many other nice properties of . For example, the ‘dual’ sequence to the one above is the more well-known
which in this case gives a direct sum decomposition .)
So it really does make sense to glue arbitrary varieties together “transversely”, even at highly singular points! I haven’t thought about what happens if you try to include nonreducedness, though.