Take a line $L$ of the first type on a smooth cubic threefold $X$ over $\mathbb C$, then its normal bundle $N\_{L|X}$ is isomorphic to $\mathcal{O}\_L\oplus \mathcal{O}\_L$. This is equivalent to say that there is a $\mathbb P^1$-family of quadric surfaces in $\mathbb P^4$ tangent to $X$ along $L$. I'm trying to write down these quadric surfaces explicitly.


Let $L=\{x\_2=x\_3=x\_4=0\}$, then up to change of coordinates, $X$ has equation


$$x\_2x\_0^2+x\_3x\_0x\_1+x\_4x\_1^2+\text{higher order terms in }x\_2,x\_3,x\_4.$$


The dual map at $p={(x\_0,x\_1)}\in L$ is $\mathcal{D}(p)=[0,0,x\_0^2,x\_0x\_1,x\_1^2]$, which determines the hyperplane $T\_{p}X$ at $p$ and we just need to find quadric surfaces containing $L$ and have tangent planes at each $p\in L$ contained in $T\_{p}X$.


I can find two of such quadric surfaces:


$$x\_4=0,~x\_2x\_0+x\_3x\_1=0,$$
$$x\_2=0,~x\_4x\_1+x\_3x\_0=0.$$


Unfortunately, the family is not a linear combination of them and I cannot find any more such quadric surface. Note that in the 1972 paper *The intermediate Jacobian of the cubic threefolds* by Clemens and Griffiths, page 309, some constructions are given in terms of equations of varieties of lines of the quadric surfaces in the Grassmannian $Gr(2,5)$. However, there seems to be a typo in the defining equations (of curve $B(\alpha\_0,\alpha\_1)$ in the paper), which I couldn't fix.


How to find the entire $\mathbb P^1$-family (hopefully in equations)? Any comments or suggestions will be appreciated!



Quadric surfaces tangent to a cubic threefold along a line of first type
It turns out that my question is all about linear algebra. As I mentioned in the chat, to find such a quadric surface, one needs to find a line $L'$ in $\mathbb P^4$, which lies in the normal direction of $L$ inside the cubic threefold $X$. Such a line is the image of a nonzero section $s$ in the normal bundle $N\_{L|X}\cong \mathcal{O}\oplus \mathcal{O}$ and the quadric surface is determined by the family of disjoint lines $\cup\_{t\in \mathbb C}ts(L)$.


In other words, by linearizing the local equation of the Fano variety of lines $F$ of $X$ in the Grassmannian $Gr(2,5)$ (i.e., throw out higher-order terms in 6.14 in Clemens-Griffiths), we can determine that the equation of $L'\_t=ts(L)$ is


$$
 \begin{cases}
 tax\_1+x\_2=0,\\
 tbx\_0-x\_4=0,\\
 tax\_0-tbx\_1-x\_3=0.
 \end{cases}\tag{1}\label{1}
 $$


with $(a,b)\in \mathbb P^1$. By standard linear algebra, we cancel out $t$ and find that the $\mathbb P^{1}$-family of quadric surfaces are given by


$$
 \begin{cases}
 \text{I}.~a^2x\_4+b^2x\_2-abx\_3=0,\\
 \text{II}.~ax\_2x\_0+ax\_3x\_1-bx\_2x\_1=0,\\
 \text{III}. ~ax\_4x\_0-bx\_4x\_1-bx\_3x\_0=0.
 \end{cases}\tag{2}\label{2}
 $$


in $\mathbb P^1\_{[a,b]}\times \mathbb P^4\_{[x\_0,...,x\_4]}$, with one linear equation and two quadric equations. Note that the family is not a complete intersection: When $a\neq 0$, the equation $\text{III}$ is redundant while when $b\neq 0$, the equation $\text{II}$ is redundant. Moreover, the points $(a,b)=(1,0)$ and $(0,1)$ correspond to the two "obvious" quadric surfaces that mentioned in the question.


By the way, I finally fixed the typo in Clemens-Griffiths p.309, about the defining equation of the curve $B(\alpha\_0,\alpha\_1)$: the fourth equation should be `u_5=\cdots=u_n=z_4=\cdots=z_n=0`. (The original paper has a `z_2` instead of `z_4`, which is wrong.) It turns out that the 1-parameter family of lines in Clemens-Griffiths is exactly the family $(\ref{1})$ we defined and $(\ref{2})$ are the explicit equations for
the $\mathbb P^1$-family of quadric surfaces tangent to $X$ along $L$ mentioned in Lemma 6.18.


