Polarized varieties without arithmetically Cohen-Macaulay bundles and section rings without graded maximal Cohen-Macaulay modules in characteristic zero
Abstract
We exhibit smooth polarized surfaces (Y,H) over C carrying no nonzero arithmetically Cohen-Macaulay bundle of any rank. Equivalently, their section rings are three-dimensional normal ℕ-graded 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 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, H2<KY2−8χ(𝒪Y)=τ(Y), the signature of Y over C, leaves no nonzero ℰ with H1(Y,ℰ(tH))=0 for all t. For Ulrich bundles, being semistable, this is Bogomolov's inequality; what is new is that no stability hypothesis and no condition on the Hilbert polynomial are needed, the Harder-Narasimhan filtration of ℰ being compared instead with the Horrocks splitting of its direct image on ℙ2. Hirzebruch's Hesse surfaces with Hn=4An−En, n≥3, qualify. In dimension m the obstruction reads (m+1)Hm≥(KX2−2c2(X))· Hm−2, the inequality Lopez proved for Ulrich bundles; products and complete intersections inside them give examples in every dimension ≥3, and the threefold examples show that the characteristic-p hypothesis in the criterion of Shimomoto-Tavanfar is essential.
Related
- Existence of Ulrich bundles on polarized surfaces and weakly Ulrich bundles on polarized varieties in positive characteristic
- On the existence of polarized varieties without weakly Ulrich bundles
- Varieties Admitting Polarized Galois Endomorphisms
- Automorphisms of hypersurfaces in polarized varieties
- Characteristic-free characterizations of projective space for smooth toric varieties
BibTeX
@misc{anghel2026polarized,
title = {{Polarized varieties without arithmetically Cohen-Macaulay bundles and section rings without graded maximal Cohen-Macaulay modules in characteristic zero}},
author = {Cristian Anghel},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.15589},
eprint = {2609.15589},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.15589},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Polarized varieties without arithmetically Cohen-Macaulay bundles and section rings without graded maximal Cohen-Macaulay modules in characteristic zero
AU - Cristian Anghel
PY - 2026
DA - 2026/09/14
PB - arXiv
DO - 10.48550/arxiv.2609.15589
UR - https://arxiv.org/abs/2609.15589
AB - We exhibit smooth polarized surfaces (Y,H) over C carrying no nonzero arithmetically Cohen-Macaulay bundle of any rank. Equivalently, their section rings are three-dimensional normal ℕ-graded 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 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<K_Y^2−8χ(𝒪_Y)=τ(Y), the signature of Y over C, leaves no nonzero ℰ with H^1(Y,ℰ(tH))=0 for all t. For Ulrich bundles, being semistable, this is Bogomolov's inequality; what is new is that no stability hypothesis and no condition on the Hilbert polynomial are needed, the Harder-Narasimhan filtration of ℰ being compared instead with the Horrocks splitting of its direct image on ℙ^2. Hirzebruch's Hesse surfaces with H_n=4A_n−E_n, n≥3, qualify. In dimension m the obstruction reads (m+1)H^m≥(K_X^2−2c_2(X))· H^m−2, the inequality Lopez proved for Ulrich bundles; products and complete intersections inside them give examples in every dimension ≥3, and the threefold examples show that the characteristic-p hypothesis in the criterion of Shimomoto-Tavanfar is essential.
ER -