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I have several questions on Hilbert scheme of Gushel-Mukai varieties. Let $X$ be a Gushel-Mukai fourfold and let $\mathcal{H}_3$ be Hilbert scheme of twisted cubics. I was wondering what is the dimension of $\mathcal{H}_3$? If I consider the locus of reducible twisted cubic, say a line and a conic. Since Hilbert scheme of lines is of dimension $3$ and Hilbert scheme of conics is of dimension $5$, then dimension of such locus is $3+5-1=7$. But I am not sure the dimension of locus of smooth twisted cubics?

The second question, if $Y$ is a Gushel-Mukai threefold, is there any space of rational curves or space of elliptic curves on $Y$ such that its dimension is $6,7,8$?

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  • $\begingroup$ The expected dimension of the Hilbert scheme is the Euler characteristic of the normal bundle, which can be easily computed. $\endgroup$
    – Sasha
    Commented Jan 6, 2022 at 19:11
  • $\begingroup$ These kinds of questions can be answered using techniques that you can find in the textbook of J'anos Koll'ar, "Rational curves on algebraic varieties". In particular, if the characteristic is zero, then "sufficiently general" lines on the fourfold are "free". Also, deformations of unions of $e$ "free lines" are "free" rational curves of degree $e$, and the locus of free rational curves of degree $e$ is smooth of the "expected" dimension $2e+1$. $\endgroup$
    – Jason Starr
    Commented Jan 7, 2022 at 19:49

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