I was trying to understand the proof of Proposition 2.1 on Huybrechets' Lectures on K3 Surfaces (see page 85).
First please let me introduce the set-up:
Let $(X,L)$ be a polarized K3 surface of degree $2d=(L)^2$ with the Hilbert polynomial $P(t)=dt^2+2$, where $L$ is am ample line bundle.
We know $L^3$ is very ample, so $X$ can be embedded into $\mathbf{P}^N$ such that $\mathcal{O}(1)|_X\simeq L^3$. Hence we have the Hilbert polynomial of $X\subset \mathbf{P}^N$ with respect to $\mathcal{O}(1)$ is $P(3t)$.
Now, we can consider the Hilbert scheme $\mathrm{Hilb}:=\mathrm{Hilb}^{P(3t)}_{\mathbf{P}^N}$ of all closed subscheme $Z\subset \mathbf{P}^N$ with Hilbert polynomial $P(3t)$.
Also, we have the universal family by $\mathcal{Z}\subset \mathrm{Hilb}\times \mathbf{P}^N$ and flat projection $\mathcal{Z}\to \mathrm{Hilb}$.
There exists a subscheme $H\subset \mathrm{Hilb}$ with the following universal property: A morphism $T\rightarrow \mathrm{Hilb}$ factors through $H\subset \mathrm{Hilb}$ if and only if the pull-back $$f:\mathcal{Z}_T\rightarrow T$$ of the universal family $\mathcal{Z}\rightarrow \mathrm{Hilb}$ satisfies:
i) The morphism $f:\mathcal{Z}_T\rightarrow T$ is a smooth family with all fibres being K3 surfaces.
ii) If $p:\mathcal{Z}_T\rightarrow \mathbf{P}^N$ is the natural projection, then $$p^*\mathcal{O}(1)\simeq L^3\otimes f^*L_0$$ for some $L\in \mathrm{Pic}(\mathcal{Z}_T)$ and $L_0\in \mathrm{Pic}(T)$.
iii) The line bundle $L$ in (ii) is primitive (i.e. not a power of any line bundle) on each geometric fiber.
iv) For all fibres $\mathcal{Z}_s$ of $f:\mathcal{Z}_T\to T$, restriction yields isomorphisms $$H^0(\mathbf{P}^N_{k(s)},\mathcal{O}(1))\xrightarrow{\sim}H^0(\mathcal{Z}_s,L_s^3).$$
The idea of the proof is to say (i) and (iv) together define an open subscheme $H'$ to which we restrict since they are open conditions. Then show that there exists a universal subscheme $H\subset H'$ defined by (ii) and (iii).
My questions are:
- Why is (iv) an open condition?
2. Why are irreducibility and vanishing of $H^1(\mathcal{O})$ open conditions?
Why do we care about irreducibility here? If we want to show the fibers are K3 surfaces, we need complete and non-singular, instead of irreducibility.
Why is the triviality of a line bundle (i.e. $\mathcal{O}\simeq \omega$) a closed condition?
5. Assume that all the above statements are true, I think we get an open subscheme of $T$, not $\mathrm{Hilb}$. It is confusing why it concludes we get a subscheme in the Hilbert scheme $\mathrm{Hilb}$? It there anything to do with the universal property?
Any answers, hints, and references are welcome!