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

TL;DR

This paper proves the Eisenbud-Goto conjecture for a specific class of projectively normal varieties with mild singularities, advancing understanding of the conjecture's validity in algebraic geometry.

Contribution

It establishes the conjecture for 2-very ample, projectively normal varieties with certain mild singularities, a previously unresolved case.

Findings

Eisenbud-Goto conjecture holds for these varieties

The result applies to varieties with factorial, rational, hypersurface, and Gorenstein singularities

Advances the classification of varieties satisfying the conjecture

Abstract

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.