Normal bundle to a double line in quartic threefold I work on complex numbers. Let Q \subset \PP^4 be a quartic threefold that contains a double line l. I want to compute the normal sheaf of l in Q, and I seem to find only two possible cases, i.e.
O_l(-1) \oplus O_l  and  O_l(-2)\oplus O_l(1)
can someone tell me which is the right one? There should be some paper by Iskovskikh about that but I am not able to find a copy of that.
 A: 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 $T_X |_\ell$.
