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arXiv · 2210.07174

Surface counterexamples to the Eisenbud-Goto conjecture

Jong In Han·Sijong Kwak
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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.

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Jong In Han, Sijong Kwak. 2022-10-13. Surface counterexamples to the Eisenbud-Goto conjecture. https://doi.org/10.1090/tran%2F9192

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