Polarized varieties without arithmetically Cohen-Macaulay bundles and section rings without graded maximal Cohen-Macaulay modules in characteristic zero
Year: 2026
Type: preprint
Abstract: We exhibit smooth polarized surfaces $(Y,H)$ over $\mathbf{C}$ carrying no nonzero arithmetically Cohen-Macaulay bundle of any rank. Equivalently, their section rings are three-dimensional normal $\mathbb{N}$-graded $\mathbf{C}$-domains, with an isolated singularity at the vertex, admitting no nonzero finitely generated graded maximal Cohen-Macaulay module. To the author's knowledge no such ring was previously known. Their existence contrasts with the theorem of Hartshorne, Hochster and Peskine-Szpiro that a three-dimensional $\mathbb{N}$-graded domain over a perfect field of characteristic $p>0$ always has one. As a consequence, in characteristic zero Hochster's small Cohen-Macaulay conjecture admits no graded refinement. The nonexistence criterion is numerical: for $|H|$ base-point-free with finite morphism, $H^2
Source: arXiv (Cornell University)
Author: Anghel, Cristian
Language: English
Subfield: Algebra and Number Theory
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W7213340067