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While studying and solving some exercises on Hilbert schemes, I've come across many problems in Hartshorne's book on deformation theory which ask the reader to show certain properties such as irreducibility, openness or non-singularity. Below I list a few examples.

(Assume $ k $ is an algebraically closed field.)

(1) In $ \operatorname{Hilb}^{3t+1} ( \mathbb{P}^3_k ) $, the twisted cubic curves form a non-singular open subset of an irreducible component $ H_0 $ of dimension $ 12 $, also courtesy of Piene and Schlessinger's paper On the Hilbert scheme compactification of the space of twisted cubics. This Hilbert scheme has another irreducible component $ H'_0 $ corresponding to plane elliptic curves union a point not on the curve.

(2) In $ \operatorname{Hilb}^8( \mathbb{P}^4_k ) $, there is an irreducible component of dimension $ 32 $ that has a non-singular open subset corresponding to $ 8 $-tuples of distinct points in $ \mathbb{P}^4_k $.

(3) The complete intersection curves in $ \mathbb{P}^3_k $ obtained by homogeneous polynomials of degrees $ a,b $ form a non-singular open subset of an irreducible component of the corresponding Hilbert scheme.

Question: How does one justify these properties, particularly the openness of certain subsets and irreducibility of components? Although these assertions seem intuitively clear, are there strategies in general that allow you to see these rigorously? For example, in the paper by Piene and Schlessinger, $ H'_0 $ is stated to be irreducible without proof and I'm not sure how to do this.

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  • $\begingroup$ I edited your question to include the name of the paper, and, while I was there, bolded the question to help it stand out. I hope that was all right. $\endgroup$
    – LSpice
    Aug 16, 2020 at 21:59
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    $\begingroup$ One can show that these spaces are irreducible by parameterizing a dense open subset. For instance, the Hilbert scheme $M$ of plane cubics in $P^3$ is parameterized by first choosing a plane, and then a cubic in that plane; so it is birational to a projective bundle over $Gr(2,4)$, and thus it is irreducible. Hence $H_0'$ is also irreducible, since it is birational to the product $M\times P^3$ (letting the extra point vary over the $P^3$). Such `set-theoretic' arguments are enough for openness and irreducibility, which are topological in nature. $\endgroup$
    – Ennio Mori cone
    Aug 16, 2020 at 22:15
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    $\begingroup$ @LSpice, thank you for the edits. $\endgroup$
    – Cranium Clamp
    Aug 18, 2020 at 16:09
  • $\begingroup$ @wnx, can you elaborate on the projective bundle part? I think you made a typo, should be $ Gr(3,4) $. If I understand correctly, that has a canonical vector bundle $ \mathcal{E} $ of rank 3 and then taking $ \mathbb{P}( \operatorname{Sym}^3 \mathcal{E}^{*} ) $ gives you a cubic? $\endgroup$
    – Cranium Clamp
    Aug 18, 2020 at 16:23
  • $\begingroup$ Yes, indeed Gr(3,4). Also, your interpretation looks correct. $\endgroup$
    – Ennio Mori cone
    Aug 19, 2020 at 22:20

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