ag.algebraic geometry - Normal bundle to a double line in quartic threefold

Let $X$ be your threefold (I assume it is smooth) and $ell subset X$ a line. We have the Gauss map $ell rightarrow mathbb{P}^2$ which associates ot a point x the tangent $T_x X$. Here the target $mathbb{P}^2$ is the space of hyperplanes containing $ell$.

Since the map is given by derivatives of an equation of $X$, it has degree $3$, so it is either 1:1 on a cubic or 3:1 on a line. In the first case the cubic spans the whole $mathbb{P}^2$, so the intersection of all tangent hyperplanes along $ell$ is $ell$ itself. In the second case there is a plane $P supset ell$ which is everywhere tangent along $ell$.

I believe if the double line is contained in $X$ we are in the second case. Now these two cases can be used to distinguish the normal of $ell$ in $X$ as follows. First we can compute by adjunction $c_1(N_{ell, X}) = -1$. Since a vector bundle on a line splits, we must have $N_{ell, X} = mathcal{O}(e_1) oplus mathcal{O}(e_2)$, with $e_1 + e_2 = -1$. Since this is a subbundle of $N_{ell, mathbb{P}^4} = bigoplus mathcal{O}(1)$, each $e_i leq 1$, leaving the two cases you mentioned, namely $(e_1, e_2) = (0, -1)$ or $(1, -2)$.

Now it comes the part where I actually did not do the computations, but it should be the same as the case of a cubic fourfold, where I have worked everything out. Namely you can distinguish the two cases according to the number of sections of $N_{ell, X}$. You have to write explicitly a generic section of $N_{ell, mathbb{P}^4}$; these form a space of dimension $6$.

Then you impose that such a section is actually tangent to $X$; this lowers the dimension by something which depends on the derivatives of $X$. If you do the computation, you will find that this dimension is different according to the two cases I have described above. I believe it turns out that $h^0(ell, N_{ell, X} = 1$ in the first case and $2$ in the second (which should be the one where the line is double).

A final remark: to perform the above computation it may be easier to observe that the normal exact sequence for the inclusion of the line into $X$ splits and work with sections of $T_X |_ell$.

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