Algebraic Surfaces of General Type with Small c²₁ in Positive Characteristic

Abstract

Abstract. We establish Noether's inequality for surfaces of general type in positive characteristic. Then we extend Enriques' and Horikawa's classification of surfaces on the Noether line, the so-called Horikawa surfaces. We construct examples for all possible numerical invariants and in arbitrary characteristic, where we need foliations and deformation techniques to handle characteristic 2. Finally, we show that Horikawa surfaces lift to characteristic zero.

Relative Noether inequality on fibered surfaces

“…See [BHPV] for char k = 0 and [Li1,Li2] for char k > 0. If the equality holds, then X has a hyperelliptic pencil (cf.…”

Geography of Irregular Gorenstein 3–folds

“…Moreover, based on several results about linear series on surfaces in positive characteristic (c.f. [26,27]), our method can be also used study the fibered 3-folds in positive characteristic. It is also interesting to compare Theorem 7.1 with the slope inequalities proved by Ohno [33] and Barja [1] for fibered 3-folds over curves.…”

The Canonical Map and Horikawa Surfaces in Positive Characteristic

“…Over the complex numbers, surfaces with K 2 = 2p g − 4 have been described by Enriques in [En,Capitolo VIII.11] and a detailed analysis of surfaces with K 2 ≤ 2p g − 3 has been carried out by Horikawa in [Hor1] and [Hor2]. In positive characteristic, the description of surfaces with K 2 = 2p g − 4 is the same as in characterstic zero [Lie2].…”

“…From the explicit description of L in Theorem 3.7 and Theorem 3.8 we get the following vanishing results, which will be important later on Proof. For even Horikawa surfaces and odd Horikawa surfaces of type B 2 this is straight forward, e.g., using [Lie2,Lemma 3.5]. For the remaining odd Horikawa surfaces, let ν : S → S be the blow-up in {x, y}.…”