1
$\begingroup$

Let $X\subset\mathbb{P}^{n+1}$ be a smooth cubic hypersurface and $p\in X$. The linear projection $\pi_p: X \dashrightarrow \mathbb{P}^n$ with center $p$ induces a double cover $\tilde{X}\to\mathbb{P}^n$ ramified at a quartic hypersurface $Y\subset\mathbb{P}^n$. Here $\tilde{X}$ is the blow up of $X$ at $p$. Now for $n\leq 2$ it follows from classical theory on elliptic curves ($n=1$) and del Pezzo surfaces ($n=2$) that in fact every smooth double cover of $\mathbb{P}^n$ ramified at a quartic hypersurface arises in that way.

Now my question is whether that remains true for $n\geq3$ as well or if there are such double covers that are not the blow up of a cubic hypersurface?

$\endgroup$

1 Answer 1

3
$\begingroup$

When $n \ge 3$ the hypersurface $Y$ obtained from $X$ is singular along an intersection of a plane, quadric, and cubic. Indeed, its equation can be written as $$ f_Y = 4f_1 f_3 - f_2^2,\tag{*} $$ where $f_1$, $f_2$, and $f_3$ are the linear, quadratic, and cubic parts of the equation $f_X$ of $X$ at $p$. Consequently, $$ Z := \{f_1 = f_2 = f_3 = 0\} \subset \operatorname{Sing}(Y). $$ In particular, a general quartic hypersurface does not correspond to a cubic. If you want to identify a quartic with a hypersurface obtained from a cubic, you need to write its equation in the form (*).

$\endgroup$

You must log in to answer this question.

Not the answer you're looking for? Browse other questions tagged .