Abstract
We investigate the vanishing of H^1(X,mathcal {O}_X(-D)) for a big and nef mathbb {Q}-Cartier mathbb {Z}-divisor D on a log del Pezzo pair (X,Delta ) over a perfect field of positive characteristic p.
Highlights
It has long been known that the (Kodaira and Kawamata–Viehweg) vanishing theorems, so fundamental to birational geometry in characteristic zero, in general fail for surfaces in positive characteristic [20]
In [6] the authors prove the existence of an integer p0 such that over an algebraically closed field of characteristic p > p0 every log del Pezzo surface satisfies Kawamata–Viehweg vanishing theorem
We show that Kodaira vanishing holds on a del Pezzo surface of bounded index I in characteristic p > p0(I ) where p0(I ) is an explicit polynomial in the index I
Summary
It has long been known that the (Kodaira and Kawamata–Viehweg) vanishing theorems, so fundamental to birational geometry in characteristic zero, in general fail for surfaces in positive characteristic [20]. In [6] the authors prove the existence of an integer p0 such that over an algebraically closed field of characteristic p > p0 every log del Pezzo surface satisfies Kawamata–Viehweg vanishing theorem. On a log del Pezzo surface over an algebraically closed field of characteristic p ≥ 4c + 1 we can prove that H 1(X , OX (K X + A)) = 0 for every ample Q-Cartier Weil divisor A of Cartier index ≤ c (see Remark 3.5). It follows from his work that Kodaira vanishing holds for log del Pezzo surfaces of Picard rank one over an algebraically closed field of characteristic p > 5 (see [6, Lemma 6.1]). This implies that the characteristic p > 5 in Theorem D is optimal
Sign-in/Register to view highlights & summary Sign-in/Register to access full text options 
Free) 
Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a call
