Surface counterexamples to the Eisenbud-Goto conjecture

TL;DR

This paper constructs the first known counterexamples to the Eisenbud-Goto conjecture for projective surfaces in , using binomial rational maps, expanding understanding of the conjecture's limitations.

Contribution

It introduces the first surface counterexamples to the Eisenbud-Goto conjecture, previously only known for higher dimensions, via binomial rational maps.

Findings

Counterexamples for projective surfaces in  are constructed.

Counterexamples challenge the universality of the Eisenbud-Goto conjecture.

Investigation of projective invariants and geometric properties of the counterexamples.

Abstract

It is well known that the Eisenbud-Goto regularity conjecture is true for arithmetically Cohen-Macaulay varieties, projective curves, smooth surfaces, smooth threefolds in $\mathbb{P}^5$, and toric varieties of codimension two. After J. McCullough and I. Peeva constructed counterexamples in 2018, it has been an interesting question to find the categories such that the Eisenbud-Goto conjecture holds. So far, surface counterexamples have not been found while counterexamples of any dimension greater or equal to 3 are known. In this paper, we construct counterexamples to the Eisenbud-Goto conjecture for projective surfaces in $\mathbb{P}^4$ and investigate projective invariants, cohomological properties, and geometric properties. The counterexamples are constructed via binomial rational maps between projective spaces.