Consider the Hilbert scheme $H$ of conics in $\mathbb{P}^3$. It is easy to see that there exists a closed subscheme $H'$ of $H$ parametrizing $2$ lines intersecting at a point. This can be seen as the image of the Hilbert flag scheme $Hilb_{P_1,P_2}$ where $P_1$ (resp. $P_2$) is the Hilbert polynomial of a line (resp. a conic) under the second projection map. Is this subscheme singular at every point i.e., is $H$ singular at every point of $H'$?

Is there a general condition when a subscheme of a Hilbert scheme (for example the image under the projection from a Flag Hilbert scheme as above) singular at every point? We know from definition that this is equivalent to saying the dimension of the global sections of the normal sheaf of the curve is higher than the dimension of the scheme. What would be interesting would be to get a condition in terms of intersection of curves.


1 Answer 1


Each conic in $P^3$ is contained in a plane $P^2 \subset P^3$, and the conic contained in a given plane are parameterized by $P^5$. Becaue of that the Hilbert scheme of conincs in $P^3$ is a $P^5$-fibration over $(P^3)^\vee = P^3$. In particular it is smooth.


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