Universal Plane Curve and Moduli Spaces of 1-dimensional Coherent Sheaves

Abstract

We show that the universal plane curve M of fixed degree d ≥ 3 can be seen as a closed subvariety in a certain Simpson moduli space of 1-dimensional sheaves on ℙ₂ contained in the stable locus. The universal singular locus of M coincides with the subvariety M′ of M consisting of sheaves that are not locally free on their support. It turns out that the blow up Bl _M′ M may be naturally seen as a compactification of M _B  = M∖M′ by vector bundles (on support).

Key Words:

• Coherent sheaves
• Simpson moduli spaces
• Vector bundles on curves

2010 Mathematics Subject Classification:

• 14D20
• 14C20

ACKNOWLEDGMENTS

The author thanks Mario Maican for his valuable comments regarding Section 3. Many thanks as well to an unknown referee for helpful comments, improvements, and suggestions.

Notes

This observation is due to an unknown referee.

Communicated by R. Piene.