An integral version of Zariski decompositions on normal surfaces

Abstract

We show that any pseudo-effective divisor on a normal surface decomposes uniquely into its "integral positive" part and "integral negative" part, which is an integral analog of Zariski decompositions. As an application, we give a generalization of the Kawamata-Viehweg vanishing, Ramanujam's 1-connected vanishing and Miyaoka's vanishing theorems on surfaces. By using this vanishing result, we give a simple proof of Reider-type theorems including the log surface case and the relative case.

Vanishing theorems and adjoint linear systems on normal surfaces in positive characteristic

“…Proof. The proof is similar to that of Proposition 3.19 in [11]. We write D ′ := π * D = π * D + D π , where D π is a π-exceptional Q-divisor on X ′ with D π = 0.…”

“…In Section 5, we study adjoint linear systems on normal surfaces in positive characteristic as an application of the vanishing theorems obtained in Section 4. The proof of the main result (Theorem 5.2) is almost similar to that of Theorem 5.2 in [11]. The only difference is to use the chain-connected component decomposition (cf.…”

“…After that, Serrano [39] and Paoletti [31] proved extension theorems for integral curves on smooth surfaces. These results were generalized in [11] for possibly singular curves on normal surfaces in characteristic 0. In this paper, as an application of the Reider-type theorem (Theorem 5.5), we give a positive characteristic analog of this extension theorem: Theorem 1.7 (Theorem 6.1).…”

“…This paper is a continuation of [11]. The purpose of this paper is to establish the vanishing theorem and the criterion for spannedness of adjoint linear systems |K X + D| for "positive" divisors D on normal surfaces X in positive characteristic.…”

“…The only difference is to use the chain-connected component decomposition (cf. Corollary 1.7 in [22]) instead of the integral Zariski decomposition (Theorem 3.5 in [11]). In Section 6, we give extension theorems for normal surfaces in positive characteristic by using the Reider-type theorem (Theorem 5.5) obtained in Section 5.…”