Pluricanonical maps of stable log surfaces

Abstract

Stable surfaces and their log analogues are the type of varieties naturally occuring as boundary points in moduli spaces. We extend classical results of Kodaira and Bombieri to this more general setting: if $(X,\Delta )$ is a stable log surface with reduced boundary (possibly empty) and $I$ is its global index, then $4I(K_X+\Delta )$ is base-point-free and $8I(K_X+\Delta )$ is very ample. These bounds can be improved under further assumptions on the singularities or invariants, for example, $5(K_X+\Delta )$ is very ample if $(X,\Delta )$ has semi-canonical singularities.

arXiv commentv2: improved section on vanishing (removing some gaps pointed out by the referee), Thm. 6.1 now treats the full (log-)canonical ring, fixed many typos; v3: fixed some typos

References in corpus (11)

• Stable pair, tropical, and log canonical compact moduli of del Pezzo surfaces
• Moduli of varieties of general type
• The canonical sheaf of Du Bois singularities
• Compact moduli of plane curves
• Moduli spaces $M_{g,n}(W)$ for surfaces
• Stable varieties with a twist
• Fundamental theorems for semi log canonical pairs
• Explicit compactifications of moduli spaces of Campedelli and Burniat surfaces
• Stable Degenerations of Surfaces Isogenous to a Product II
• Geography of Gorenstein stable log surfaces
• Singular curves and their compactified Jacobians

Cited by in corpus (5)

• Log-canonical pairs and Gorenstein stable surfaces with $K_X^2=1$
• Geography of Gorenstein stable log surfaces
• Effective basepoint-free theorem for semi-log canonical surfaces
• Computing invariants of semi-log-canonical surfaces
• Virus infections, Corona surfaces, and extra components in the moduli space of stable surfaces