The Eisenbud-Goto conjecture for projectively normal varieties with mild singularities

For a nondegenerate projective variety $X$, the Eisenbud-Goto conjecture asserts that $\mathrm{reg}X\le \mathrm{deg}X-\mathrm{codim}X+1$. Despite the existence of counterexamples, identifying the classes of varieties for which the conjecture holds remains a major open problem. In this paper, we prove that the Eisenbud-Goto conjecture holds for $2$-very ample projectively normal varieties with factorial, rational, hypersurface singularities and isolated Gorenstein singularities.