It is known that a Hilbert scheme of degree $d$ curves in $\mathbb P^3$ can have dimension more than $4d$. But, does it imply that for some types of curves there are such a curve through any, say, $2d+1$ points? (or $3d$)
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asked Mar 15, 2014 at 18:08
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Take complete intersections of degree $(4,4)$: there is a pencil of quartics passing through $2\times 16+1=33$ general points. More generally, complete intersections of degree $(d,d)$ will pass through any $n(d).d$ points, with $n(d)\rightarrow\infty$ with $d$.
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$\begingroup$ oh, good. And why complete intersection is generally irreducible? $\endgroup$ Commented Mar 16, 2014 at 8:20
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$\begingroup$ Take a general (4,4) complete intersection $C$,it is smooth and irreducible, and take 33 general points on it. It is easy to see that $C$ is the unique such curve through these points. $\endgroup$– abxCommented Mar 16, 2014 at 8:25