Pencil of plane curves

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.

It is a pencil whose members are plane curves.

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.

This is the degree-three case of a Fermat equation; in general $x^n+y^n+z^n=0$ defines a Fermat curve.

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.