Generalized logarithmic sheaf on smooth projective surfaces

TL;DR

This paper introduces generalized logarithmic sheaves on smooth projective surfaces, explores their Torelli and stability properties, and describes their moduli spaces, with particular focus on blow-ups of the projective plane and cubic surfaces.

Contribution

It defines generalized logarithmic sheaves on surfaces and investigates their Torelli, stability, and moduli space properties, extending classical concepts to new geometric settings.

Findings

Torelli property studied for generalized logarithmic sheaves

Stability conditions for these sheaves established

Moduli spaces explicitly described for specific surfaces

Abstract

We define the notion of generalized logarithmic sheaves on a smooth projective surface, associated to a pair consisting of a reduced curve and some fixed points on it. We then set up the study of the Torelli property in this setting, focusing mostly in the case of the blow-up of the projective plane on a reduced set of points and, in particular, in the case of the cubic surface. We also study the stability property of generalized logarithmic sheaves as well as carrying out the description of their moduli spaces.