A cubic hypersurface in $\mathbb{P}^{4}$ that passes through $7$ points in general position with multiplicity $2$.

Question

I am reading Rick Miranda's "Linear systems of plane curves". A cubic hypersurface in $\mathbb{P}^{4}$ that passes through $7$ points in general position with multiplicity $2$ is not expected to exist. Nevertheless, it exists, and I am trying to understand Miranda's example.

There exists a rational normal curve $X$ that passes through $7$ points in general position in $\mathbb{P}^{4}$. Let $S(X)$ be its secant variety. $S(X)$ is supposed to be the variety we are looking for.

The secant variety of an irreducible curve is three-dimensional unless the curve is contained in a plane. A rational normal curve in $\mathbb{P}^{n}$ spans $\mathbb{P}^{n}$, so it is not contain in a plane. According to this, dim$(S(X))=3$.

Now, why is the degree of $S(X)$ equal to $3$? I think it suffices to show that any $l\in\mathbb{G}(1,4)-S(X)$ satisfies $\#X\cap l=3$, but I am not able to prove it.

Now, let's suppose that $S(X)$ passes through any of the seven points $p$ with multiplicity $1$. Then dim $T_{p}S(X)=3$. Since $X$ spans $\mathbb{P}^{4}$, there are secant lines to $X$ that contain $p$ in $4$ independent directions. I am not able to understand why this fact contradicts dim $T_{p}S(X)=3$. Is $\{l\in\mathbb{G}(1,4):\text{l is secant to $X$ and $p\in l$}\}\subseteq T_{p}S(X)$?

Any help would be appreciated.

Answer 1

From [BO]

The case $n = 4, d = 3, k = 7$ is more subtle. In this case, since ${7\choose 3} = 7 \cdot 5$, it is expected that no cubics exist with seven given singular points. But indeed through seven points there is a rational normal curve $C_4$, which, in a convenient system of coordinates, has equation $\operatorname{rk} \begin{pmatrix}x_0 &x_1& x_2\\ x_1 &x_2& x_3\\ x_2 &x_3& x_4\end{pmatrix} \leq 1$. Its secant variety is the cubic with equation $\operatorname{det} \begin{pmatrix}x_0 &x_1 &x_2\\ x_1 &x_2 &x_3\\ x_2 &x_3 &x_4\end{pmatrix} = 0$, which is singular along the whole $C_4$. This is the same J invariant which describes harmonic 4-ples on the projective line. The paper [CH] contains a readable proof of the uniqueness of the cubic singular along $C_4$.

[BO] Maria Chiara Brambilla and Giorgio Ottaviani, On the Alexander-Hirschowitz theorem, J. Pure Appl. Algebra 212 (2008)

[CH] C. Ciliberto, A. Hirschowitz, Hypercubiques de P4 avec sept points singuliers génériques. C. R. Acad. Sci. Paris Sér. I Math. 313 (1991), no. 3, 135-137