Smooth Counterexamples to the Eisenbud--Schreyer--Weyman Ulrich Existence Problem

Technical Summary: Smooth Counterexamples to the Eisenbud–Schreyer–Weyman Ulrich Existence Problem

Problem Statement
The paper addresses the Eisenbud–Schreyer–Weyman (ESW) existence problem, which asks whether every embedded projective variety $X \subset \mathbb{P}^N$ carries an Ulrich sheaf. The author adopts the standard scheme-theoretic formulation: an Ulrich sheaf on $X$ is a coherent $\mathcal{O}_X$-module with full support that is Ulrich with respect to the hyperplane class $\mathcal{O}_X(1)$. On smooth surfaces, such sheaves are locally free (vector bundles).

While Coskun and Huizenga proved that for any smooth surface $X$ and ample divisor $H$, the pair $(X, mH)$ carries Ulrich bundles for sufficiently large $m$, the ESW question is sharper: for a fixed very ample class $H$ (i.e., a fixed projective embedding), must an Ulrich object exist? The paper investigates whether this fixed-embedding formulation admits counterexamples among smooth complex projective surfaces.

Methodology and Numerical Obstruction
The core of the paper's argument relies on a numerical obstruction derived from the Bogomolov–Hodge inequality, which holds regardless of the Picard rank of the surface.

1. The Inequality: Let $(S, H)$ be a smooth polarized surface. If $E$ is a rank-$r$ Ulrich bundle on $S$, the author defines a shifted class $\alpha = c_1(E) - \frac{r}{2}(K_S + 3H)$. Using the Riemann–Roch theorem, semistability, Bogomolov's inequality ($\Delta(E) \ge 0$), and the Hodge index theorem ($\alpha^2 \le 0$), the paper establishes the necessary condition:
   $$H^2 \ge K_S^2 - 8\chi(\mathcal{O}_S) = \sigma(S)$$
   where $\sigma(S)$ is the topological signature of the underlying four-manifold.

   This inequality refines a previous observation by Beauville (2017). Beauville's search for counterexamples relied on the assumption that the Picard rank is one ($\text{rk NS}(S)=1$), which forces $\alpha \equiv 0$ and yields the same inequality. The author demonstrates that the hypothesis of Picard rank one is unnecessary; the term $\alpha^2 \le 0$ ensures the inequality holds for any Picard rank. Thus, any smooth polarized surface with $H^2 < \sigma(S)$ cannot carry an Ulrich bundle of any rank.

2. Geometric Construction: To construct surfaces violating this inequality, the author utilizes Hirzebruch–Kummer covers associated with line arrangements in $\mathbb{P}^2$.

   • Let $\mathcal{A}$ be an arrangement of $k$ lines.
   • Let $X_n$ be the exponent-$n$ Kummer cover of $\mathbb{P}^2$ branched over $\mathcal{A}$.
   • Let $Y_n$ be the minimal resolution of the singularities of $X_n$.
   • The author employs a Rees-algebra/Segre mechanism to identify very ample classes on $Y_n$. Specifically, if the singular ideal of $X_n$ is generated by monomials of degree $\nu$, then the class $H_n = (\nu+1)A_n - E_n$ is very ample, where $A_n$ is the pullback of the hyperplane class and $E_n$ is the exceptional divisor.

Key Results and Counterexamples
The paper constructs explicit infinite families of smooth counterexamples where $H_n^2 < \sigma(Y_n)$.

1. The Hesse Family (Main Counterexample):

   • Based on the classical Hesse arrangement of 12 lines (9 fourfold points, 12 double points).
   • The singular ideal is generated by cubic monomials ($\nu=3$), yielding the very ample class $H_n = 4A_n - E_n$.
   • Theorem 1.1: For $n=3$, the surface $Y_3$ with polarization $H_3 = 4A_3 - E_3$ satisfies:
     $$H_3^2 = 7 \cdot 3^9 < 16 \cdot 3^9 = \sigma(Y_3)$$
     Consequently, $uc(Y_3, H_3) = \infty$; no Ulrich sheaf exists for this embedding.
   • Theorem 1.2: This extends to all $n \ge 3$. The surfaces $Y_n$ are pairwise non-isomorphic, and the ratio $H_n^2 / \sigma(Y_n) \to 0$ as $n \to \infty$.
   • Geometric Significance: For $n=3$, $Y_3$ is a compact ball quotient (satisfying $K_{Y_3} = 3L$). The polarization $H_3$ is constructed by moving "off the ray" from the canonical root $L = K/3$ via $H = L - A$. While $L^2 = \sigma(Y_3)$, the subtraction of the nef class $A$ reduces the self-intersection strictly below the signature threshold while preserving very ampleness.

2. Companion Families:

   • Dual Hesse Arrangement: Yields counterexamples for all $n \ge 5$ with polarization $H_n = 6A_n - E_n$.
   • Complete Quadrangle: Yields counterexamples for all $n \ge 6$ with polarization $H_n = 3A_n - E_n$. At $n=5$, this family realizes the numerical boundary $K^2 = 3H^2 = \sigma$ in higher Picard rank.

3. General Mechanism (Theorem 1.3):
   The paper establishes a general criterion (Theorem 6.1) for any line arrangement admitting a degree-$\nu$ generation datum. If the arrangement parameters satisfy specific growth conditions (specifically $Q_{\mathcal{A}} > 0$), the resulting family $(Y_n, H_n)$ provides counterexamples for all sufficiently large $n$.

Significance and Claims
The paper claims to provide the first known smooth polarized varieties where Ulrich bundles of every rank are excluded.

• Resolution of the Fixed-Embedding Question: The results demonstrate that the answer to the ESW problem is negative for smooth complex projective surfaces when the embedding is fixed.
• Removal of Picard Rank Restriction: The work shows that the numerical obstruction $H^2 \ge \sigma(S)$ is rank-independent, correcting the previous assumption that counterexamples required Picard rank one.
• Chow Form Implications: By the ESW construction, the non-existence of an Ulrich sheaf implies that the Chow form of the embedded surface admits no linear determinantal representation arising from an Ulrich sheaf.
• Systematic Phenomenon: The counterexamples are not isolated anomalies but part of infinite families of non-isomorphic surfaces. The violation of the Ulrich threshold becomes arbitrarily large in relative terms because the signature $\sigma(Y_n)$ grows quadratically with $n$ (specifically as $n^{k-1}$ in the leading term of the normalized formula), while the polarization square $H_n^2$ grows only as $n^{k-3}$.

The paper concludes that while asymptotic existence holds for $(X, mH)$, the specific geometry of the Hesse ball quotient and its Kummer relatives provides a robust obstruction to the existence of Ulrich sheaves for fixed, very ample polarizations.