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

TL;DR

This work addresses the Eisenbud–Goto conjecture by establishing the bound $\operatorname{reg} X \le \deg X - \operatorname{codim} X + 1$ for nondegenerate varieties that are $2$-very ample, projectively normal, and possess mild singularities (factorial rational singularities with $\dim T_pX \le \dim X + 1$). The authors develop vanishing theorems for singular varieties, extend the birational double point formula to factorial points, and leverage positivity of the double point divisor to bound the regularity of $\mathcal{O}_X$, which, together with projective normality, yields the desired bound on $\mathcal{I}_X$. The main theorem thus broadens the class of varieties for which the Eisenbud–Goto conjecture is known to hold, and clarifies the role of inner projections and singularities in controlling regularity. These results highlight how geometric operations on projections, combined with cohomological vanishing and divisor positivity, can enforce sharp regularity bounds in the presence of mild singularities.

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 mild singularities.

Theorems & Definitions (31)

• Conjecture 1.1: The Eisenbud--Goto conjecture, MR741934
• Theorem 1
• Definition 2.1
• Proposition 2.2: MR597077
• Theorem 2.3: MR141668, also cf. MR2095471
• Theorem 2.4: MR2645037
• Theorem 2.5: MR3217694
• Definition 2.6
• Lemma 2.7: MR3217694
• Theorem 2.8: MR3217694