Surface counterexamples to the Eisenbud-Goto conjecture

2022/10/13 by Han, Jong In, Kwak, Sijong
#Algebraic Geometry (math.AG) #FOS: Mathematics #Primary 14N05 #Secondary 13D02

paper · doi:10.48550/arxiv.2210.07174

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 ℙ⁵, 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 ℙ⁴ and investigate projective invariants, cohomological properties, and geometric properties. The counterexamples are constructed via binomial rational maps between projective spaces.

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