Reconstruction of singularities on orbifold del Pezzo surfaces from their Hilbert series

Abstract

The Hilbert series of a polarized algebraic variety (X, D) is a powerful invariant that, while it captures some features of the geometry of (X, D) precisely, often cannot recover much information about its singular locus. This work explores the extent to which the Hilbert series of an orbifold del Pezzo surface fails to pin down its singular locus, which provides nonexistence results describing when there are no orbifold del Pezzo surfaces with a given Hilbert series, supplies bounds on the number of singularities on such surfaces, and has applications to the combinatorics of lattice polytopes in the toric case.

Keywords:

• del Pezzo surface
• Hilbert series
• mirror symmetry
• orbifold singularity

2010 Mathematics Subject Classification:

• 14J45 (Primary); 14J17, 13A02 (Secondary)

Acknowledgments

The author would like to thank Al Kasprzyk for his support during this project and his collaboration in our related joint work. The author is also grateful to Alessio Corti, Mohammed Akhtar, Tom Coates, Bernd Sturmfels, and Vivek Shende for many valuable conversations.

Additional information

Funding

This research was partially funded by a grant from the London Mathematical Society.