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

Abstract

Let $X$ be a smooth projective surface with $p_g=0$, and let $H$ be an ample divisor with $h^0(\mathcal{O}_X(H))\neq0$, $\chi(\mathcal{O}_X(H))\ge q$, and $h^1(\mathcal{O}_X(H))\neq0$. We prove that no rank two bundle $\mathcal{E}$ with $c_1(\mathcal{E})=3H+K_X$, with the Ulrich value of $c_2(\mathcal{E})$, and satisfying $h^0(\mathcal{E}(-H))=0$, can arise from an extension $0 \to \mathcal{O}_X(H+K_X) \to \mathcal{E} \to \mathcal{O}_X(2H)\otimes\mathcal{I}_Z \to 0$. Thus, for this natural Cayley-Bacharach construction, non-speciality of the polarization is necessary rather than merely convenient. We then study primary Burniat surfaces. We show that every ample and base point free divisor is non-special; consequently, every polarization carries a stable special Ulrich bundle of rank two, and the surface is strictly Ulrich wild with respect to every polarization. We also locate the special ample classes on three numerical rays through $K_X$, compute explicit families on these rays, and analyze a twisted-kernel variant of the construction. The degree bound underlying the non-speciality result overlaps with recent work of Y. Cho, while the global consequences and the no-go theorem are independent.