Projective hypersurface ($X_d$)

A degree-$d$ projective hypersurface $X_d\subseteq {\mathbb{P}}_k^n$ is the closed subscheme cut out by one nonzero homogeneous polynomial of degree $d$.

For a reduced projective hypersurface $X=V(f)$ over an algebraically closed field,The Euler homogeneous function theorem makes the intersection redundant only when the characteristic does not divide the degree. In characteristic $p$, the irreducible polynomial $x_0^p+x_1^{p-1}{x}_2$ has zero gradient at $[1:0:0]$, which is not on $X$.

Multiplication by the defining polynomial of a degree-$d$ projective hypersurface gives the short exact sequenceThus the ideal sheaf of $X_d$ is ${\mathcal{O}}_{{\mathbb{P}}^n}(-d)$.

For a projective algebraic set $X\subseteq {\mathbb{P}}^n$, its affine cone is the union in ${\mathbb{A}}^{n+1}$ of the lines represented by the points of $X$, together with the origin.

The affine variety $X=V(xy-zw)\subset {\mathbb{A}}^4$ has dimension three and a unique singular point of an algebraic variety at the origin. The planes $V_X(x,z)$ and $V_X(y,w)$ have dimension two but intersect only at that origin. This is a basic failure of intersection dimension estimates at singular points.

On $Y=\{xy=zw\}\setminus \{0\}$, the formulasglue to a morphism $Y\to {\mathbb{P}}^1$. The inverse image of $[0:1]$ is the ruling plane $V(x,z)\cap Y$, so its divisor line bundle is the pullback of ${\mathcal{O}}_{{\mathbb{P}}^1}(1)$.

Given homogeneous polynomials $F,G$ of the same degree, their pencil is the one-parameter familyIts total space is the projective hypersurface $V(sF+tG)\subseteq {\mathbb{P}}^2\times {\mathbb{P}}^1$, and projection to $[s:t]$ has these plane curves as its fibres.

The Fermat cubic $x^3+y^3+z^3=0$ is a smooth projective curve: its three first partial derivatives vanish simultaneously only at the forbidden zero vector. The genus of a smooth plane curve formula gives genus one.

The pencil generated by the Fermat cubic curve and $xyz=0$ contains both a smooth genus one curve and the union of the three coordinate lines. Its total space in ${\mathbb{P}}^2\times {\mathbb{P}}^1$ is irreducible because the two generating cubics have no common factor.

If $X=V_{+}(F)\subseteq {\mathbb{P}}^n$, its affine cone is $Y=V(F)\subseteq {\mathbb{A}}^{n+1}$. If $X$ is smooth, every nonzero point of $Y$ is smooth; hence the vertex is the only possible singular point. The vertex is smooth for linear $F$ and singular whenever $\mathrm{deg}F\ge 2$.

Over an algebraically closed field of characteristic other than two, the rank-$r+1$ quadricis an irreducible $r$-dimensional cone for $r\ge 2$, and its singular locus is the omitted-coordinate vertex $[0:\cdots :0:1]$.

For $n>k\ge 0$, take an $(n-k)$-dimensional irreducible projective variety with one singular point and form its product with ${\mathbb{P}}^k$. The Segre embedding makes the product projective, and its singular locus is the singular point times ${\mathbb{P}}^k$, of dimension $k$. A quadric cone works in dimension at least two and a cuspidal cubic works in dimension one.