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!

  • 2
    $\begingroup$ How do you conclude that there is a $\mathbb{P}^1$-family of quadric surfaces in $\mathbb{P}^4$ tangent to $X$ along $L$? $\endgroup$ – Sasha Nov 1 '20 at 8:21
  • $\begingroup$ Dear @Sasha, as the normal bundle $N_{L|X}$ is $\mathcal{O}_L\oplus \mathcal{O}_L$, the image of any nonzero section $s$ is a line in $\mathbb P^4$ which is disjoint from $L$ and union of family of disjoint lines $\cup_{0\le t\le 1}ts(L)$ is Zariski dense in a quadric surface, whose tangent spaces along $L$ are contained in $TX$. Two sections determine the same quadric surface if and only if one is a multiple to the other, so there is a $\mathbb PH^0(N_{L|X})\cong \mathbb P^1$-family of such quadric surfaces. Actually, I can find these equations now. I'll write a solution below. $\endgroup$ – AG learner Nov 1 '20 at 19:29

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.


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