A no-go theorem for special Ulrich bundles, with a complement on primary Burniat surfaces

Technical Summary: A No-Go Theorem for Special Ulrich Bundles, with a Complement on Primary Burniat Surfaces

Problem Statement
The paper investigates the existence of rank two Ulrich bundles on smooth projective surfaces $X$ with geometric genus $p_g = 0$. Specifically, it addresses the "special" case where the first Chern class satisfies $c_1(E) = 3H + K_X$ in $\text{Pic}(X)$, rather than merely numerically. A central question in the literature is whether the standard Cayley–Bacharach construction (an extension of the form $0 \to \mathcal{O}_X(H+K_X) \to E \to \mathcal{O}_X(2H) \otimes \mathcal{I}_Z \to 0$) can produce such bundles when the polarization $H$ is special (i.e., $h^1(\mathcal{O}_X(H)) \neq 0$). While non-speciality ($h^1=0$) is a standard hypothesis in existence theorems (e.g., Casnati [Ca17], Beauville [Be16b]), this paper asks whether it is a necessary condition for this specific construction or merely a convenient one.

Methodology
The authors employ a combination of cohomological analysis, extension theory, and the geometry of specific surface classes:

1. Extension Criteria: The paper utilizes the Serre–Griffiths–Harris correspondence, which links the existence of a locally free extension to the Cayley–Bacharach (CB) property of a 0-dimensional subscheme $Z$ with respect to a linear system.
2. Cohomological Tension: The core argument involves analyzing the tension between the CB property (which requires $Z$ to impose dependent conditions on a linear system to ensure local freeness) and the Ulrich vanishing condition $h^0(E(-H)) = 0$ (which requires $Z$ to impose independent conditions).
3. Generic vs. Special Mechanisms: The authors contrast the "generic" mechanism of Li–Qin [LQ94], which relies on generic 0-cycles and requires non-speciality, with the "points-on-a-curve" mechanism of [An94], which places $Z$ on an irreducible curve. They demonstrate that even the latter fails for the adjoint kernel when $H$ is special.
4. Burniat Surface Analysis: For the positive results, the authors perform explicit computations on primary Burniat surfaces (minimal surfaces of general type with $p_g=q=0$ and $K_X^2=6$). They utilize the bidouble cover structure $\pi: X \to Y$ (where $Y$ is a del Pezzo surface of degree 6) to reduce cohomology calculations to the rational surface $Y$, employing eigensheaf decompositions and lattice theory.

Key Contributions and Results

1. The No-Go Theorem (Theorem 1.3)
The primary negative result establishes that for a surface with $p_g=0$ and an ample divisor $H$ with $h^1(\mathcal{O}_X(H)) \neq 0$ (and $\chi(\mathcal{O}_X(H)) \ge q$), no rank two vector bundle $E$ with $c_1(E) = 3H + K_X$ and the correct $c_2(E)$ can arise from the adjoint-kernel extension:
$$0 \to \mathcal{O}_X(H + K_X) \to E \to \mathcal{O}_X(2H) \otimes \mathcal{I}_Z \to 0$$
satisfying $h^0(E(-H)) = 0$.

• Mechanism: The proof shows that the Ulrich condition forces the length of $Z$ to be $\ell(Z) = \chi(\mathcal{O}_X(H)) + \chi(\mathcal{O}_X)$. The vanishing $h^0(E(-H))=0$ implies $h^0(\mathcal{O}_X(H) \otimes \mathcal{I}_Z) \le q$. However, the CB property (required for local freeness) implies that $Z$ imposes dependent conditions, leading to a contradiction unless $h^1(\mathcal{O}_X(H)) = 0$.
• Significance: This proves that non-speciality is a necessary hypothesis for the natural adjoint-kernel construction, not just a convenient assumption. It rules out the possibility that "special-position" mechanisms (like placing points on curves) can bypass the non-speciality requirement for this specific bundle shape.

2. Existence on Primary Burniat Surfaces (Theorem 1.5)
The authors apply their results to primary Burniat surfaces.

• Global Non-Speciality: They prove that every ample and base-point-free divisor on a primary Burniat surface is non-special ($h^1(\mathcal{O}_X(H)) = 0$). This is derived from a degree bound: an ample base-point-free divisor must have intersection number $\ge 2$ with the six elliptic ramification curves of the surface. Since $K_X$ has intersection 1 with these curves, $H - K_X$ is ample, and Kodaira vanishing applies.
• Consequences: Consequently, every polarization on a primary Burniat surface admits a $\mu_H$-stable special Ulrich bundle of rank two. Furthermore, these surfaces are "strictly Ulrich wild" with respect to every polarization, carrying simple Ulrich bundles of every even rank $r \ge 2$ in families of arbitrarily large dimension.

3. Classification of Special Ample Classes
While no polarization (very ample divisor) is special on these surfaces, the authors classify the "special ample classes" (ample but not base-point-free).

• They show that all special ample classes lie on three specific numerical rays through $K_X$.
• They compute the cohomology along these rays, showing that classes $H_u = K_X + \pi^*\mathcal{O}_Y(u(e_0 - e_1))$ are special for $u \ge 2$.
• Crucially, they prove that no class on these rays is base-point-free, explaining why they do not contradict the non-speciality of polarizations.

4. Twisted Kernels and Obstructions
The paper investigates whether "twisted" kernels (e.g., $0 \to \mathcal{O}_X(H+K_X-\epsilon) \to E \to \mathcal{O}_X(2H+\epsilon) \otimes \mathcal{I}_Z \to 0$) can salvage the construction for special polarizations.

• On primary Burniat surfaces, they analyze a specific twisted variant involving ramification curves.
• They demonstrate that even with twisting, the construction fails for the specific special classes identified. The "budget" of cohomology (the dimension of the target of the connecting map) is insufficient to absorb the speciality of the polarization while maintaining the required vanishing conditions.

5. Higher Rank and Wildness
Building on the existence of rank two bundles, the authors use the Faenzi–Pons-Llopis criterion to construct families of simple Ulrich bundles of higher even ranks. They establish that primary Burniat surfaces are strictly Ulrich wild, improving upon previous results by providing explicit dimension counts and simplicity (rather than just indecomposability).

Significance and Claims
The paper claims two main contributions:

1. A Negative Result: It clarifies the limitations of the Cayley–Bacharach construction, proving that the non-speciality of the polarization is an intrinsic requirement for the adjoint-kernel shape, not an artifact of the proof technique. This resolves a gap in the literature where the necessity of this hypothesis was not explicitly recorded.
2. A Positive Result on Burniat Surfaces: It provides a complete picture of Ulrich bundles on primary Burniat surfaces. It confirms that despite the surface's complexity, the geometry forces all polarizations to be non-special, thereby guaranteeing the existence of stable special Ulrich bundles for every polarization. This contrasts with recent work by Cho [Cho26], which established non-speciality for specific classes (like $3K_X$) but did not formulate the global result for all polarizations or the strict wildness consequences.

The authors note that their results on Burniat surfaces overlap with Cho's work regarding the degree bound on elliptic curves, but their derivation is independent and leads to stronger global conclusions. They also highlight that while their elementary methods (Cayley–Bacharach) fail for certain special cases (like the quintic surface in $\mathbb{P}^3$), the bundles likely still exist, suggesting the obstruction is positional rather than numerical.