Abstract
We show that over any algebraically closed field of positive characteristic, there exists a smooth rational surface which violates Kawamata–Viehweg vanishing.
Highlights
It is a well-known fact that Kodaira vanishing fails in positive characteristic [23]
It has often been believed that a stronger version, namely Kawamata– Viehweg vanishing, holds over a smooth rational surface
In [2], the authors and Witaszek showed that Kawamata–Vieweg vanishing holds for klt del Pezzo surfaces in large characteristic
Summary
It is a well-known fact that Kodaira vanishing fails in positive characteristic [23]. Lp2+p+1 are pairwise disjoint, we can contract all these curves and obtain a birational morphism g : X → Y onto a klt surface Y such that ρ(Y ) = 1 (cf Lemma 2.4). After Raynaud constructed the first counter-example to Kodaira vanishing in positive characteristic [23], several other people studied this problem (e.g. see [3,4,6], [15, Section 2.6], [21,26]). In [2], the authors and Witaszek showed that Kawamata–Vieweg vanishing holds for klt del Pezzo surfaces in large characteristic. If p = 2, the surface mentioned above is a smooth weak del Pezzo surface (cf Lemma 2.4), our result cannot be extended to characteristic two (see Proposition 7.1)
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