On Del Pezzo Fibrations in Positive Characteristic

Abstract

We establish two results on three-dimensional del Pezzo fibrations in positive characteristic. First, we give an explicit bound for torsion index of relatively torsion line bundles. Second, we show the existence of purely inseparable sections with explicit bounded degree. To prove these results, we study log del Pezzo surfaces defined over imperfect fields.

On the boundedness of globally F-split varieties

“…Then Sect. 4 establishes key results about the behaviour of conic bundles in sufficiently high characteristic. Next Sect.…”

“…Lemma 5. 4 We use the notation of Theorem 5.1. Suppose Z is a surface and there is t such that (G, (1+t) G ) is klt.…”

The maximal unipotent finite quotient, unusual torsion in Fano threefolds, and exceptional Enriques surfaces

“…Then the algebraic group G = Pic τ Y /k is smooth, and there are cases with H 1 (Y , O Y ) 0. Tanaka constructed Mori fiber spaces in characteristic p ≤ 3, where the generic fiber is a normal projective surfaces Y with h 0 (O Y ) = 1 having only Q-factorial klt-terminal singularities, the Q-divisor K Y is anti-ample, yet the Picard group contains elements of order p ([Tan16, Theorem 1.2], with further investigation in [BT20]). From Theorem 3.3, we see that such elements must come from the component Pic 0 Y /k of the origin.…”

“…Moreover, Tanaka [Tan16] constructed Mori fibrations X → B on threefolds in characteristic p = 2, 3 where the generic fiber Y = X η is a normal del Pezzo surface whose Picard group contains elements of order p = 2, 3. Very recently, Bernasconi and Tanaka [BT20] provided effective bounds for the torsion on del Pezzo-type surfaces over imperfect fields.…”