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

For a nondegenerate projective variety $X$, the Eisenbud-Goto conjecture asserts that $\operatorname{reg}X\leq\operatorname{deg}X-\operatorname{codim}X+1$. 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.

💡 Research Summary

The paper addresses the long‑standing Eisenbud–Goto conjecture, which predicts that for a non‑degenerate projective variety (X\subset\mathbb P^{r}) the Castelnuovo–Mumford regularity satisfies \