# Smooth hypersurface of degree $p$ in characteristic $p$

Over an algebraically closed field of characteristic $p$, does there exist a smooth degree $p$ hypersurface $X\subset\mathbb{P}^r$ that contains a plane $\Gamma$ of dimension $\lfloor \frac{r-1}{2}\rfloor$?

What about a smooth degree $p$ hypersurface $X$ such that a hyperplane section $X\cap H$ is a cone (considered as a subscheme of $H\cong \mathbb{P}^{r-1}$)?

(If we don't allow the degree to be divisible by $p$, then the Fermat hypersurface is an example of both.)

• The $27$ lines should work for $p=3$. Jan 29, 2017 at 5:21

Let me give an example for your first question for $p=2$ and $r=3$, hoping I fully understand what you are looking for.
Consider in $\mathbb{P}^3$ the hypersurface defined by $z^2+zt+xu$ in characteristic $2$. By checking the partial derivatives after localizing in every chart, it is smooth.
Now to the plane: note that if you localize at $t=1$, you get the spectrum of $k[z,x,u]/(z^2+z+xu)$. If you now intersect with any $u=\alpha$ hyperplane, then you get the spectrum of $k[z,x]/(z^2+z+\alpha x)$ which is just the most basic Artin-Schreier cover of $\mathbb{A}^1$, and is known to be again (isomorph to) an affine line.
• I see. At least for $p=2$ and odd dimension, we can go off of what you said and use $x_0x_1+x_2x_3+\cdots+x_{r-1}x_r$ as the hypersurface. Setting $x_0=0$, we see it's a cone, and setting $x_i=0$ for $i$ even, we see it contains planes of high dimension.
• Do you mean $x_i=1$ for $i$ even? In any case, it works very well indeed. What I had in mind was to take a good AS cover (i.e. isomorphic to the $n$-dimensional plane) and "homogenize" its defining equation (the philosophy being that every finite cover is projective and hence this should give a meaninful object in the projective space) in a clever way such that it gives a smooth hypersurface. Do you need me to elaborate this further in higher char or were you searching only for the (un)existence of such examples? Feb 1, 2017 at 8:34