  McCullough and Peeva found counterexamples to the Eisenbud-Goto conjecture on
the Castelnuovo-Mumford regularity by using Rees-like algebras. In this paper,
we suggest another method to construct counterexamples to the conjecture and
show that for nondegenerate projective varieties of a fixed dimension $n\geq 3$
and a fixed codimension $e\geq2$, the regularity can not be bounded above by
any polynomial function of the degree whose order is about half of the
dimension $n$. Our strategy is an unprojection process and utilizes the
possible complexity of homogeneous ideals with three generators and
exceptional-fundamental schemes. Also, our counterexamples exhibit how the
singularity affects the regularity.
