Affine hypersurface

The zero set of a nonconstant polynomial in $n$ affine variables has dimension $n-1$. Passing to the square-free part gives its radical principal ideal.

Let $k$ be algebraically closed of characteristic $p>0$ and let $f\in k[x_1,\dots ,x_n]$ be irreducible. Some partial derivative of $f$ is nonzero: otherwise every exponent is divisible by $p$, and perfection of $k$ would make $f$ a $p$th power. If the gradient vanished at every point of $Z(f)$, the Strong Hilbert Nullstellensatz would put each partial derivative in $(f)$, impossible by degree. Hence $Z(f)$ has a smooth point and minimum tangent dimension $n-1$.

Let $f$ be homogeneous of positive degree $d$ over a field whose characteristic does not divide $d$. The Euler homogeneous function theorem givesOn the level set $f=c\ne 0$, the gradient cannot vanish, so $Z(f-c)$ is a smooth hypersurface.

At a smooth point of a hypersurface, restriction to the tangent hyperplane has zero linear term. The natural hyperplane intersection therefore has tangent dimension one larger than its expected dimension and is singular at that point.

An irreducible affine plane cubic has at most one singular point. If two existed, restriction of its cubic polynomial to their joining line would have a zero of multiplicity at least two at each point. A polynomial of degree at most three cannot have those four zeros unless it vanishes identically, making the line a component and contradicting irreducibility.

The affine plane curve $y^2=z^3$ is an irreducible cuspidal cubic. Away from the origin its tangent line is $2ydy-3z^{2}dz=0$; at the origin both partial derivatives vanish, so its Zariski tangent space is the whole ambient plane and the origin is singular.

The algebraic set in ${\mathbb{A}}^3$ defined byis the disjoint union of the cuspidal cubic $V(x,y^2-z^3)$ and the two reduced points $(-1,0,0)$ and $(1,0,-1)$. Its radical vanishing ideal is the intersection of the three corresponding prime ideals.

The affine surfaceis birational to ${\mathbb{A}}^2$ and has singular locus $\{(1,0,z):z\in k\}$, an irreducible affine line.