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 $H^2\ge K_S^2-8χ(\mathcal O_S)$, 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 $H^2=7\cdot3^9<16\cdot3^9=K_Y^2-8χ(\mathcal O_Y)$, 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 $n\ge3$ the Hesse pair $(Y_n,4A_n-E_n)$ is a counterexample, the surfaces are pairwise non-isomorphic, and $H_n^2/σ(Y_n)=7/(3n^2-11)\to0$, while $K_{Y_n}^2/χ(\mathcal O_{Y_n})\to60/7\approx8.5714$. A general arrangement-theoretic Rees-algebra/Segre mechanism yields further infinite families.