# Stability of line bundles on reducible varieties

## Atoshi Chowdhury (UC Berkeley)

Over a family of varieties in which some fibers are singular (for example, reducible), the relative Picard stack (the moduli space of line bundles) may fail to be compact. I'll discuss an asymptotic stability condition for line bundles on reducible varieties, which is aimed at compactifying such relative Picard stacks. This stability condition generalizes the one arising from geometric invariant theory which was used by Caporaso in the 1990s to compactify relative Picard schemes over families of curves.
I'll present some results on counting semistable line bundles on reducible varieties of arbitrary dimension with ample or anti-ample canonical bundle, as well as similar results over degenerate K3 surfaces. I'll also discuss what remains to be investigated before these results can be used to construct proper moduli spaces of semistable line bundles.

Tuesday, September 29st, at 4:00pm in HBH 227

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