# Double covers and blow ups of cubic hypersurfaces

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?

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 (*).