Let X⊆ P^r be a nondegenerate projective variety. Its Castelnuovo–Mumford regularity is denoted by operatornamereg X, its degree by deg X, and its codimension by operatornamecodim X. Eisenbud–Goto regularity conjecture. Then operatornamereg X≤deg X-operatornamecodim X+1. This conjecture proposes a bound for the homological complexity of a projective variety in terms of its degree and codimension. It is refuted: counterexamples were found by McCullough and Peeva.

Eisenbud–Goto regularity conjecture

About 16 years old · traced to 2010

Let $X\subseteq\mathbb P^r$ be a nondegenerate projective variety. Its Castelnuovo–Mumford regularity is denoted by $\operatorname{reg} X$ , its degree by $\deg X$ , and its codimension by $\operatorname{codim} X$ . Eisenbud–Goto regularity conjecture. Then

\operatorname{reg} X\leq\deg X-\operatorname{codim} X+1.

This conjecture proposes a bound for the homological complexity of a projective variety in terms of its degree and codimension. It is refuted: counterexamples were found by McCullough and Peeva.

References

Primary source

Junho Choe, “Castelnuovo-Mumford regularity of unprojections and the Eisenbud-Goto regularity conjecture”, arXiv:2206.06151 (2024).

Additional references

6 papers in this index state this conjecture (2010–2022). The statement above is taken from the most recent of them; the others are arXiv:2106.12667, arXiv:1406.7404, arXiv:1402.1906, arXiv:1110.2124, arXiv:1012.5329.

castelnuovo-mumford-regularity commutative-algebra projective-geometry

Posted 1 month ago · edited 4 weeks ago

Progress summary

Refreshed 4 weeks ago

Claimed solved

The conjecture is false: McCullough and Peeva constructed varieties whose regularity is far larger than the proposed degree bound, while important special cases remain valid.

Proposed in 1984, the conjecture asserts the bound $\operatorname{reg}X\leq\deg X-\operatorname{codim}X+1$ for every nondegenerate projective variety $X$ . McCullough and Peeva’s constructions refute this general statement.

Known results

The bound holds for projective curves.
It holds for arithmetically Cohen–Macaulay varieties.
It holds for projective toric varieties of codimension $2$ .
It is known for smooth surfaces and smooth threefolds in $\mathbb{P}^{5}$ .

2018 counterexamples and June 2022 quantitative refinements

McCullough and Peeva produced nondegenerate prime ideals whose regularity is not bounded by any polynomial in degree; one example has $\deg X=31$ and $\operatorname{reg}X=38$ . A later construction gives, for fixed dimension $n\geq3$ and codimension $e\geq2$ , families with $\operatorname{reg}X=\Omega((\deg X)^k)$ , where $k=\lfloor(n+1)/2\rfloor$ .

Current status (as of September 2026): The general conjecture is settled negatively by the reported McCullough–Peeva counterexamples, while the sharp boundary of valid special cases remains open.

Sources

No solutions have been posted yet.