Uniruled surfaces of general type

Abstract

ABSTRACT. We give a systematic construction of uniruled surfaces in positive characteristic. Using this construction, we find surfaces of general type with non-trivial global vector fields, surfaces with arbitrarily non-reduced Picard schemes as well as surfaces with inseparable canonical maps. In particular, we show that some previously known pathologies are not sporadic but exist in abundance. CONTENTS

Algebraic Surfaces of General Type with Small c21 in Positive Characteristic

“…More precisely, the quotient X ′ := F d /η has ℓ singular points of type D 8 and m singular points of type D 12 , where we use [Lie,Proposition 2.3] to determine the singularities. Using the results of [Lie,Section 4] we see that X ′ has an ample canonical sheaf if g ≥ d + 4, i.e., X ′ is the canonical model of a surface of general type. We find p g = 2m − 2 + 2d ≥ 2d − 2 and that X ′ is a Horikawa surface.…”

“…, cf. also [Lie,Section 7]. This construction yields all examples of Horikawa surfaces with all possible values of g and d where both are even integers.…”

“…This construction yields all examples of Horikawa surfaces with all possible values of g and d where both are even integers. If 5d − 2ℓ + 2m − 2 = 4 then the quotient X ′ := F d /η acquires an elliptic singularity of type (19) 0 at the point lying below P := {x 2 = t 2 = 0} by [Lie,Proposition 2.3]. The other singular points of X ′ are again Du Val singularities of type D 8 and D 12 .…”

Lectures on Supersingular K3 Surfaces and the Crystalline Torelli Theorem

“…If k = C, then complex conjugation induces the Hodge symmetry h 1,0 = h 0,1 . However, in positive characteristic, this Hodge symmetry may fail in general (see [Se58] and [Li08] for examples), and thus, we have to compute h 1,0 (X) another way: using the isomorphism E ∨ ∼ = E ⊗det(E), which holds for locally free sheaves of rank 2 (see [Har77, Exercise II.5.16], for example), we find T X ∼ = Ω 1 X/k for a K3 surface, and thus…”

Counterexamples of Kodaira vanishing for smooth surfaces of general type in positive characteristic

“…In light of the few characteristic-free results on pluricanonical systems and rank 2 vector bundles on surfaces not of general type (see [E88,ShB91a]), the current focus is on surfaces of general type. Yet little is known on the classification of general type surfaces with certain positive characteristic pathologies (see [L08,LS09]). For this matter, we would like to further understand the known examples of Kodaira non-vanishing and to come up with new ones.…”