Smooth Counterexamples to the Eisenbud--Schreyer--Weyman Ulrich Existence Problem
Abstract
In their 2003 article in the Journal of the American Mathematical Society, Eisenbud, Schreyer and Weyman asked whether every embedded projective variety carries an Ulrich sheaf. In 2017 Beauville proposed a numerical route toward a surface with no Ulrich bundles: in Picard rank one, existence forces H2≥ KS2−8χ(𝒪S), suggesting a search near the Bogomolov--Miyaoka--Yau boundary. We show that the Picard-rank-one hypothesis is not needed for the obstruction: a rank-independent Bogomolov--Hodge argument gives the same inequality on every smooth polarized surface. Using additional Neron--Severi directions on Hirzebruch--Kummer resolutions, the exponent-$3$ Hesse surface admits a very ample class H=4A−E with H2=7·39<16·39=KY2−8χ(𝒪Y), hence a smooth counterexample to the Eisenbud--Schreyer--Weyman problem in its standard formulation. Consequently its Chow form has no ESW-type linear determinantal representation arising from an Ulrich sheaf on the embedded surface. Moreover, for every n≥3 the Hesse pair (Yn,4An−En) is a counterexample, the surfaces are pairwise non-isomorphic, and Hn2/σ(Yn)=7/(3n2−11)→0, while KYn2/χ(𝒪Yn)→60/7≈8.5714. A general arrangement-theoretic Rees-algebra/Segre mechanism yields further infinite families.
Related
- The recent counterexamples to the Eisenbud-Schreyer-Weyman Ulrich problem (2003) and to Hochster's small Cohen-Macaulay conjecture (1973): a survey, with a generalization to arbitrary dimension
- A simple construction of smooth counterexamples to the Loewner and Carathéodory conjectures
- On the existence of smooth varieties of any dimension carrying no Ulrich bundles for some very ample line bundle
- On the existence of nonadmissible irreducible smooth representations of p-adic GLn
- Existence of multi-solitons for logarithmic Schrödinger equation with a perturbation of a smooth L2-subcritical function nonlinearity
BibTeX
@misc{anghel2026smooth,
title = {{Smooth Counterexamples to the Eisenbud--Schreyer--Weyman Ulrich Existence Problem}},
author = {Cristian Anghel},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.02718},
eprint = {2609.02718},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.02718},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Smooth Counterexamples to the Eisenbud--Schreyer--Weyman Ulrich Existence Problem
AU - Cristian Anghel
PY - 2026
DA - 2026/09/02
PB - arXiv
DO - 10.48550/arxiv.2609.02718
UR - https://arxiv.org/abs/2609.02718
AB - In their 2003 article in the Journal of the American Mathematical Society, Eisenbud, Schreyer and Weyman asked whether every embedded projective variety carries an Ulrich sheaf. In 2017 Beauville proposed a numerical route toward a surface with no Ulrich bundles: in Picard rank one, existence forces H^2≥ K_S^2−8χ(𝒪_S), suggesting a search near the Bogomolov--Miyaoka--Yau boundary. We show that the Picard-rank-one hypothesis is not needed for the obstruction: a rank-independent Bogomolov--Hodge argument gives the same inequality on every smooth polarized surface. Using additional Neron--Severi directions on Hirzebruch--Kummer resolutions, the exponent-$3$ Hesse surface admits a very ample class H=4A−E with H^2=7·3^9<16·3^9=K_Y^2−8χ(𝒪_Y), hence a smooth counterexample to the Eisenbud--Schreyer--Weyman problem in its standard formulation. Consequently its Chow form has no ESW-type linear determinantal representation arising from an Ulrich sheaf on the embedded surface. Moreover, for every n≥3 the Hesse pair (Y_n,4A_n−E_n) is a counterexample, the surfaces are pairwise non-isomorphic, and H_n^2/σ(Y_n)=7/(3n^2−11)→0, while K_Y_n^2/χ(𝒪_Y_n)→60/7≈8.5714. A general arrangement-theoretic Rees-algebra/Segre mechanism yields further infinite families.
ER -