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Let $n\geq 4$ and $X\subset \mathbb{P}^n$ be a smooth hypersurface. Let $p(t)\in \mathbb{Q}$ be a polynomial and we consider the Hilbert scheme $Hilb^{p(t)}(X)$.

If a point $[Y]\in Hilb^{p(t)}(X)$ corresponds to a complete intersection subscheme $Y\subset X$, what can we say about the local property of the Hilbert scheme near $[Y]$? For example, is the smoothness or normality of $Hilb^{p(t)}(X)$ at the point $[Y]$ already known?

Thanks for any answer and references.

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  • $\begingroup$ Is $Y$ the intersection of $X$ with a complete intersection in projective space whose dimension is one greater than the dimension of $Y$? $\endgroup$
    – Jason Starr
    Oct 15, 2022 at 11:36
  • $\begingroup$ @JasonStarr Yes, here I assume that $Y$ is a complete intersection of $X$ with some other hypersurfaces in $\mathbb{P}^n$. $\endgroup$
    – Kim
    Oct 15, 2022 at 13:23
  • $\begingroup$ Then, yes, smoothness is known. For LCI closed subschemes of a smooth scheme, the conormal sheaf is locally free, and the Ext^1 group of this sheaf equals H^1 of the dual locally free sheaf (the "normal bundle"). The normal bundle of the complete intersection in $X$ equals the restriction of the normal bundle of the "ambient" complete intersection in $\mathbb{P}^n$. This is a direct sum of positive degree line bundles. Now use the computation of cohomology of line bundles on $\mathbb{P}^n$ and chasing through Koszul complexes. $\endgroup$
    – Jason Starr
    Oct 16, 2022 at 10:56
  • $\begingroup$ @JasonStarr Thanks for your detailed answer! Is there any reference for your argument, so I can cite it in my notes? $\endgroup$
    – Kim
    Oct 16, 2022 at 11:09
  • $\begingroup$ I am sure there is a better reference, but one source that I know is the thesis of Yoon-Ho (Alex) Lee. I will try to find the link . . . $\endgroup$
    – Jason Starr
    Oct 16, 2022 at 11:45

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