Prime Ideals and Three-generated Ideals with Large Regularity

McCullough, Jason

Comptes Rendus. Mathématique, Tome 362 (2024) no. G3, pp. 251-255

Ananyan and Hochster proved the existence of a function $\Phi (m,d)$ such that any graded ideal $I$ generated by $m$ forms of degree at most $d$ in a standard graded polynomial ring satisfies $\mathrm{reg}\left(I\right)\le \Phi (m,d)$. Relatedly, Caviglia et. al. proved the existence of a function $\Psi \left(e\right)$ such that any nondegenerate prime ideal $P$ of degree $e$ in a standard graded polynomial ring over an algebraically closed field satisfies $\mathrm{reg}\left(P\right)\le \Psi (deg(P\left)\right)$. We provide a construction showing that both $\Phi (3,d)$ and $\Psi \left(e\right)$ must be at least doubly exponential in $d$ and $e$, respectively. Previously known lower bounds were merely super-polynomial in both cases.

Classification : 13D02, 13D05, 13P20

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