Past exam of the mathematics course of the University of Cambridge / 2009 / iii / Paper 21 / 3 / b / Solution

Take the projective hypersurfaceAs a polynomial in $X_0$ over $\mathbb{C}[X_1,X_2,X_3]$, it is primitive because $X_1$ and $X_2^2$ are coprime. It is linear and irreducible over the fraction field, so Gauss lemma for polynomials proves irreducibility. Thus it is an irreducible surface.

Its gradient is $(X_1,X_0,-2X_2,0)$. The simultaneous zero set on $S$ is exactlyThe Jacobian criterion proves every other point is smooth. At that vertex the affine equation in the chart $X_3=1$ has no linear term, so the tangent space has dimension three whereas the surface has dimension two; the vertex is genuinely singular. This is a projective quadric cone with one singular vertex.