Irreducibility of the singular locus of a cubic hypersurface

Let $Z\subseteq \mathbb{P}^{N}$ be an irreducible cubic hypersurface, i.e. $Z=V(F)$ for certain homogeneous irreducible polynomial $F\in K[X_{0},\ldots,X_{N}]$ of degree $3$. Let us suppose that its singular locus $$X=V(\frac{\partial F}{\partial X_{0}},\ldots,\frac{\partial F}{\partial X_{N}})$$ is smooth (in particular $Z$ is not a cone) and that the secant variety of $X$ is $SX=Z$. I would like to prove that in this case $X$ is irreducible.

I don't know if this is true, but at least, I would like to know if there is some easy criterion to check irreducibility under these hypothesis.

• One idea is to try to prove that the polynomial $F$ is homoloidal, i.e. the rational map $\nabla F: \mathbb P^N \dashrightarrow \mathbb P^N$ defined by the partial derivatives of $F$ is birational. Once this is done you can try to apply the classification of homoloidal cubic polynomials due to Etingof, Kazhdan, Polishchuck carried out in the paper 'When is the Fourier transform of an elementary function elementary?' (arxiv.org/abs/math/0003009v2) in order to conclude. Feb 25 '16 at 18:20

1 Answer

Edit. I missed the condition that the secant variety should span the hypersurface. I am leaving the example below for any case. I will think about the secant condition.

Original Post. That is not true, with the possible exception of a few finite fields such as $\mathbb{F}_2$. Already for $N\geq 3$, the zero scheme of the following degree $3$ polynomial is integral, normal, and has singular locus equal to $\{[1,0,0,\dots,0],[0,1,0,\dots,0]\}$. For a hypersurface in $\mathbb{P}^N$, if the singular locus has codimension $\geq 3$ in $\mathbb{P}^N$, then automatically the hypersurface is integral and normal. This follows from Serre's Criterion for normality. $$F(X_0,X_1,X_2,X_3,X_4\dots,X_N) = X_0X_1(X_2+X_3) +$$ $$(a_3X_2+a_2X_3)X_2X_3 + G(X_4,\dots,X_N).$$ Here $a_2$ and $a_3$ are distinct nonzero elements of the field (these exist if the field is not $\mathbb{F}_2$), and $G(X_4,\dots,X_N)$ is a homogeneous degree $3$ polynomial whose critical locus in $\mathbb{A}^{N-3}$ is just the origin, e.g., $X_4^3+\dots+X_N^3$ if the characteristic is not $3$ (there are similar examples in characteristic $3$ except for finitely many finite fields $\mathbb{F}_{3^r}$).

• Thank you! Just in case it helps, the Veronese surface (and every Severi variety) is a variety in the conditions of the question. In addition (and if I am not wrong), since $X$ is smooth, irreducibility is equivalent to connectedness. Feb 23 '16 at 16:41