We classify complex K3 surfaces of zero entropy admitting an elliptic fibration with only irreducible fibers. These surfaces are characterized by the fact that they admit a unique elliptic fibration with infinite automorphism group. We furnish an explicit list of 30 Néron-Severi lattices corresponding to such surfaces. Incidentally, we are able to decide which of these 30 classes of surfaces admit a unique elliptic pencil. Finally, we prove that all K3 surfaces with Picard rank ≥ 19 and infinite automorphism group have positive entropy.
We study the moduli spaces of elliptic K3 surfaces of Picard number at least 3, that is, U ⊕ −2k -polarized K3 surfaces. Such moduli spaces are proved to be of general type for k 220. The proof relies on the low-weight cusp form trick developed by Gritsenko, Hulek and Sankaran. Furthermore, explicit geometric constructions of some elliptic K3 surfaces lead to the unirationality of these moduli spaces for k < 11 and for 19 other isolated values up to k = 64.
We classify all non-extendable 3-sequences of half-fibers on Enriques surfaces. If the characteristic is different from 2, we prove in particular that every Enriques surface admits a 4-sequence, which implies that every Enriques surface is the minimal desingularization of an Enriques sextic, and that every Enriques surface is birational to a Castelnuovo quintic.
CONTENTS1. Introduction 1 2. c-sequences 4 3. Special 3-sequences 6 4. Non-extendable 3-sequences 13 4.1. Examples 13 4.2. Special non-extendable 3-sequences 17 4.3. Non-special non-extendable 3-sequences 19 5. Projective models of Enriques surfaces 22 References 23
We define the over-exceptional lattice of a minimal algebraic surface of Kodaira dimension
0
0
. Bounding the rank of this object, we prove that a conjecture by Campana [Ann. Inst. Fourier (Grenoble) 54 (2004), pp. 499–630] and Corvaja–Zannier [Math. Z. 286 (2017), pp. 579–602] holds for Enriques surfaces, as well as K3 surfaces of Picard rank
≥
6
\geq 6
apart from a finite list of geometric Picard lattices.
Concretely, we prove that such surfaces over finitely generated fields of characteristic
0
0
satisfy the weak Hilbert Property after a finite field extension of the base field. The degree of the field extension can be uniformly bounded.
scite is a Brooklyn-based organization that helps researchers better discover and understand research articles through Smart Citations–citations that display the context of the citation and describe whether the article provides supporting or contrasting evidence. scite is used by students and researchers from around the world and is funded in part by the National Science Foundation and the National Institute on Drug Abuse of the National Institutes of Health.