On non-vanishing of cohomologies of generalized Raynaud polarized surfaces

We consider a family of slightly extended version of Raynaud’s surfaces XX over the field of positive characteristic with Mumford–Szpiro type polarizations ZZ, which have Kodaira non-vanishing H1(X,Z−n)≠0H1(X,Z−n)≠0 for all 1≤n≤N1≤n≤N with some N≥1N≥1. The surfaces are at least normal but smooth under a special condition. We also give a fairly large family of non-Mumford–Szpiro type polarizations Za,bZa,b with Kodaira non-vanishing.