Hilbert scheme of points and passing curves

It is well known that through five points on a projective plane you can pass a conic.

Q. What happens when points collide ?

More precisely: if I consider a more simple question of two points and passing a line through them, then if I take a point in a Hilbert scheme of two points $Hilb^2$ I can pass exactly one line when this points collide.

Q Now, can I draw a conic (maybe singular) if I choose a point in $Hilb^5$ of projective plane? Or may be I need some different version of moduli space of points? The symmetric powers not work at all.

• "Now, can I draw a unique conic (maybe singular) if I choos a point in $\text{Hilb}^5$ of projective plane." No, you cannot find a unique conic, even in the nice case that the point parameterizes $5$ distinct, reduced points. For instance, consider the case when $4$ of the $5$ points are collinear. – Jason Starr Dec 23 '17 at 16:45
• So maybe the question should be rephrased as - given a general point of the diagonal, is there a unique conic passing through it ? – aginensky Dec 23 '17 at 16:57