  Classification of AS-regular algebras is one of the most important projects
in noncommutative algebraic geometry. Recently, Itaba and the first author gave
a complete list of defining relations of $3$-dimensional quadratic AS-regular
algebras by using the notion of geometric algebra and twisted superpotential.
In this paper, we extend the notion of geometric algebra to cubic algebras, and
give a geometric condition for isomorphism and graded Morita equivalence. One
of the main results is a complete list of defining relations of $3$-dimensional
cubic AS-regular algebras corresponding to $\mathbb{P}^1 \times \mathbb{P}^1$
or a union of irreducible divisors of bidegree $(1,1)$ in $\mathbb{P}^1 \times
\mathbb{P}^1$. Moreover, we classify them up to isomorphism and up to graded
Morita equivalence in terms of their defining relations.
