A remarkable class of elliptic surfaces of amplitude 1 in weighted projective space

TL;DR

This paper explores four classes of elliptic surfaces of amplitude 1 in weighted projective spaces, constructing their moduli, analyzing monodromy and period maps, and applying findings to moduli spaces of specific surfaces, with computational support.

Contribution

It introduces four classes of elliptic surfaces in weighted projective spaces, constructs their moduli spaces, and investigates their monodromy, period maps, and Torelli-type properties, extending understanding of these surfaces.

Findings

Moduli spaces are constructed for four classes of elliptic surfaces.

Monodromy and period maps are determined and found to be non-injective.

A Torelli-type theorem is established for the properly elliptic classes.

Abstract

Surfaces of amplitude 1 in ordinary projective space are of general type, but this need not be the case in weighted projective spaces. Indeed, there are 4 classes of quasi-smooth weighted hypersurfaces in $\mathbf{P}(1,2,a,b)$ of amplitude 1 with an elliptic pencil cut out by hyperplanes. Their moduli spaces are constructed, the monodromy of their universal families is determined as well as their period maps. These all turn out to be non-injective. We analyse the reason behind this, which for each type is different. For the two classes that give properly elliptic surfaces this leads to a mixed Torelli-type theorem as in the case of the Catanese-Kunev-Todorov surfaces. We added an application to certain compactifications of moduli spaces of surfaces of general type with $K^2=1$, $p_g=2$ and $q=0$, as well as detailed SageMath-calculations. The appendix written by Wim Nijgh shows that the general member of the type 1 and type 2 elliptic family has "trivial" Picard lattice, i.e. is spanned by fiber components and a multisection.