My problem concerns the construction of Mumford's famous example of an Hilbert scheme that has a non-reduced component. Let $k$ be an algebraically closed field, and let $X\subset \mathbf{P}^{3}_{k}$ be a regular cubic surface.
Let $H$ be the hyperplane section of $X$, and $L$ any line. It can be proved that there exists a divisor $C\in |4H+2L|$ such that the corresponding curve is irreducible.
For any cubic surface $X$, let $W_{X}$ be the set of all divisors $C\in |4H+2L|$ such that the corresponding curves are irreducible, for all lines $L\subset X$. Consider the family $W=\cup_{X} W_{X}$, obtained as the union of all $W_{X}$, for all regular cubic surfaces $X\subset \mathbf{P}^{3}_{k}$. I would like to prove the following two facts:
(i) W is irreducible;
(ii) dim($W$)$=19+$dim($C$).
Bibliographical references are also welcome. Please note that I'm still a beginner with the algebraic geometry. I apologise in advance if the problem is poorly stated, or if it is just a special case of something more general.
