Surfaces of amplitude 1 in the ordinary projective space are of the general type, but this need not be the case in weighted projective spaces. Indeed, there are 4 classes of quasi-smooth weighted hypersurfaces in P(1, 2, a, b) of amplitude 1 with an elliptic pencil cut out by hyperplanes. Their moduli spaces are constructed, and the monodromy of their universal families is determined as well as their period maps which turn out to be generally immersive. For those that are not, a mixed Torelli theorem holds. We added an application to certain compactifications of moduli spaces of surfaces of the general type with K2 = 1, pg = 2 and q = 0 as a follow up of Gallardo et al. (2022), as well as detailed SageMath-calculations. The appendix written by Wim Nijgh shows that the general member of the types 1 and 2 elliptic family has a “trivial” Picard lattice, i.e., is spanned by fiber components and a multisection.
A remarkable class of elliptic surfaces of amplitude 1 in weighted projective spaces
Gregory Pearlstein
Co-primo
Writing – Original Draft Preparation
;
2024-01-01
Abstract
Surfaces of amplitude 1 in the ordinary projective space are of the general type, but this need not be the case in weighted projective spaces. Indeed, there are 4 classes of quasi-smooth weighted hypersurfaces in P(1, 2, a, b) of amplitude 1 with an elliptic pencil cut out by hyperplanes. Their moduli spaces are constructed, and the monodromy of their universal families is determined as well as their period maps which turn out to be generally immersive. For those that are not, a mixed Torelli theorem holds. We added an application to certain compactifications of moduli spaces of surfaces of the general type with K2 = 1, pg = 2 and q = 0 as a follow up of Gallardo et al. (2022), as well as detailed SageMath-calculations. The appendix written by Wim Nijgh shows that the general member of the types 1 and 2 elliptic family has a “trivial” Picard lattice, i.e., is spanned by fiber components and a multisection.I documenti in IRIS sono protetti da copyright e tutti i diritti sono riservati, salvo diversa indicazione.
