Eisenbud–Goto regularity conjecture

Let $X\subseteq\mathbb P^r$ be a nondegenerate projective variety. Its Castelnuovo–Mumford regularity is denoted by $\operatorname{reg} X$, its degree by $\deg X$, and its codimension by $\operatorname{codim} X$. Eisenbud–Goto regularity conjecture. Then

$$\operatorname{reg} X\leq\deg X-\operatorname{codim} X+1.$$

This conjecture proposes a bound for the homological complexity of a projective variety in terms of its degree and codimension. It is refuted: counterexamples were found by McCullough and Peeva.