Effective bounds on singular surfaces in positive characteristic

Abstract

Abstract. Using the theory of Frobenius singularities, we show that 13mKX`45mA is very ample for an ample Cartier divisor A on a Kawamata log terminal surface X with Gorenstein index m, defined over an algebraically closed field of characteristic p ą 5.

Bogomolov-Sommese vanishing and liftability for surface pairs in positive characteristic

“…Now, by [55, Theorem 1.2], it follows that 13mK X + 45mA X is very ample. Moreover, we can see that (13m − 3)K X + (45m − 14)A X is nef and hence H i (X, O X (13mK X + 45mA X )) = 0 for all i > 0 by [55,Proposition 6.5]. We set H X := 13mK X + 45mA X .…”

On the Kodaira vanishing theorem for log del Pezzo surfaces in positive characteristic

“…Proof of Theorem A By base change to an algebraic closure of k we can assume that k is algebraically closed. By [24] the divisor (13 − 45I )I K X is very ample. By [13,Theorem 1.3] there is an inequality K 2 X ≤ max{9, 2I + 4 + 2 I }.…”

“…Proof Let H = 13m K X − 45m 2 (K X + ). According to [24], H is a very ample Cartier divisor. By [24,Lemma 6.2], we have inequalities (K X + )•K X ≥ 0 and K 2 X ≤ 128m 5 (2m− 1).…”

“…According to [24], H is a very ample Cartier divisor. By [24,Lemma 6.2], we have inequalities (K X + )•K X ≥ 0 and K 2 X ≤ 128m 5 (2m− 1). By [13,Theorem 1.3], we have an inequality (K X + ) 2 ≤ 2m + 4 + 2 m .…”

On log del Pezzo surfaces in large characteristic

“…There exist positive integers and which satisfy the following property (see e.g. [Wit15, Corollary 1.4 and Remark 6.3]): for every over as in the lemma, we can find a very ample divisor on such that:…”