Castelnuovo–Mumford Regularity of Unprojections and the Eisenbud–Goto Regularity Conjecture

McCullough and Peeva found sequences of counterexamples to the Eisenbud–Goto conjecture on the Castelnuovo–Mumford regularity by using Rees-like algebras, where entries of each sequence have increasing dimensions and codimensions. In this paper we suggest another method to construct counterexamples to the conjecture with any fixed dimension $n\geq 3$ and any fixed codimension $e\geq 2$. Our strategy is an unprojection process and utilizes the possible complexity of homogeneous ideals with three generators. Furthermore, our counterexamples exhibit how singularities affect the Castelnuovo–Mumford regularity.