Past exam of the mathematics course of the University of Cambridge / 2018 / ii / Paper 2 / 24I / b / Solution

Write a polynomial $f\in W$ by its coefficients. Evaluation of $f$ and of each first partial derivative at $p$ is polynomial in these coefficients and in the coordinates of $p$. Consequently the bad incidence locusis Zariski closed, and $U=(W\times k^3)\setminus B$ is Zariski open. Here a hypersurface is regarded as smooth at an ambient point outside it automatically.

In fact $B$ is empty. For homogeneous $f$ of degree $d>0$, the Euler homogeneous function theorem saysIf $(f,p)\in B$, its left side is zero while its right side is $d$, which is nonzero because $\mathrm{char}k=0$. Thus every nonzero level is covered by the smoothness of a nonzero homogeneous level set, andwhich is certainly nonempty and Zariski open.