Strong non-vanishing of cohomologies and strong non-freeness of adjoint line bundles on $n$-Raynaud surfaces

We begin by formally defining n-Tango curves and n-Raynaud surfaces. Our investigation then focuses on the pathological behaviors exhibited by n-Raynaud surfaces. As a direct corollary of this analysis, we present a concise disproof of Fujita’s conjecture for surfaces in positive characteristics.