Incidence Correspondence

Question

A useful tool in Algebraic Geometry is the incidence correspondence. Loosely speaking, it is a set of the form $$\{(p,X): p \text{ a fixed dimension subscheme of } Y \text{ and } X \text{ a specific type of subscheme} \}.$$ For example one could consider the incidence correspondence of lines in $\mathbb{P}^2$ with a point on them, or cubics in $\mathbb{P}^3$ with a line on them.

It is not too hard to see that in each of the previous cases, the resulting scheme is a variety by writing down explicit equations in coordinates. In the first case the variety lives in $\mathbb{P}^2\times \mathbb{P}^2$ and with a bit of work it is not too hard to show it is a projective bundle over $\mathbb{P}^2.$ The second case is a subvariety of $\mathbb{P}^{19}\times G(1,3)$.

My question is the following. Is there a way to get both of the previous examples in a more natural way then explicitly writing down equations? Should I even expect there to be one? Generally when working with incidence correspondences one is interested in properties such as smoothness and irreducibility and most authors I have seen conclude these from the equations. Since the first case above does end up being a projective bundle, I would really hope for there to be a natural way to construct it.

Thanks.

Answer 1

There's several issues here.

(1) Is a given incidence correspondence actually a closed variety?

(2) What are explicit equations for the correspondence in the product of the relevant spaces?

(3) What are geometric properties of the incidence correspondence?

Most often, questions (1) and (3) are studied and little attention is paid to (2).

For (1), things can be generalized a fair bit. For instance, suppose $X$ is a variety, $H$ is an ample divisor, and $P$, $Q$ are two Hilbert polynomials with respect to the ample divisor $X$. Then there are (projective) Hilbert schemes $Hilb_P(X)$ and $Hilb_Q(X)$ parameterizing closed subschemes of $X$ with Hilbert polynomials $P$ and $Q$. Then there are a couple different natural incidence correspondences, for example

$\{(Z,Z'):Z\subset Z'\} \subset Hilb_P(X)\times Hilb_Q(X)$

$\{(Z,Z'):Z\cap Z'\neq \emptyset\}\subset Hilb_P(X)\times Hilb_Q(X),$

and it is easy to verify that these conditions are closed (although the correct scheme structure may be less clear). One can instead restrict attention to a closed subvariety of the Hilbert scheme (so as to not use all the components of the Hilbert scheme, for instance, in case the geometric objects you care about are not entirely determined by their Hilbert polynomials). It is also easy to generalize to cases with more factors. These types of constructions mean that arguments for the closedness of incidence correspondences are almost never written down, as anything reasonable that you can write down will be closed so long as the families of objects under consideration are themselves closed.

In practice, (2) is rarely of any theoretical interest, unless these are very special varieties. Perhaps there is a large algebraic group acting and the ideal can be studied via representation theory, or perhaps the variety has very small dimension or is otherwise very simple, in which case some information might be learned from the ideal.

Regarding (3), geometric properties of an incidence correspondence are only very rarely (roughly in the same cases as discussed for (2)) determined by studying the defining equations. Most often, the reason we study an incidence correspondence $\Sigma \subset X\times Y$ is because we have some question about one of the projection maps, say $\Sigma \to X$; for instance, we may wonder if it is dominant. We then ideally answer this by studying the other projection, which ideally is easier to understand. Likewise, properties like dimension and irreducibility are hopefully easily understood by studying the projections, and smoothness can sometimes be analyzed as well (although smoothness is often not so important in basic applications).