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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 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.
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Incidence Correspondence

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.