It is well known that a smooth cubic surface has $27$ distinct lines. Explicitly, if we choose a planar representation, i.e., blowup $\mathbb P^2$ at $6$ general points $p_1,...,p_6$, the $27$ lines are (1) $E_i$, $1\le i\le 6$, the exceptional divisors, (2) $F_{ij}$, $1\le i<j\le 6$, the proper transform of lines joining $p_i$ and $p_j$, and (3) $Q_i$, $1\le i\le 6$, the proper transform of conics passing $5$ points except $p_i$.

When a cubic surface acquires with one node ($A_1$ singularity), it has $21$ lines. One can think this happens in a specialization as the $6$ points become to lie on a single conic, and the line $E_i$ and $Q_i$ coincide in the limit as a double line, for $i=1,...,6$, while the rest of the $15$ lines $F_{ij}$ stays simple. So $27$ is interpreted as $2\times 6+15$.

What happens in general? My understanding is that, since the number $27$ (or $2875$ for quintic threefolds) is calculated via the intersection theory, it should be interpreted as the *length of the Hilbert scheme of lines*, especially when the cubic surface is not too singular and the number of lines is still finite.

According to Dolgachev's book section 9.2.2, *all cubic surfaces with at worst rational double point singularities have finitely many lines*. (e.g., a cubic surface with an $A_2$ singularity has $15$ lines; a cubic surface with an $E_6$ singularity has only $1$ line.)

So my question is, **is there work been done to describe the Hilbert scheme of lines for cubic surfaces with rational double point singularities, or is there a geometric interpretation of how the number $27$ are attributed to the multiplicities of geometric lines in those cubic surfaces?**