Minimal Log Discrepancy and Orbifold Curves

Abstract

We show that the minimal log discrepancy of any isolated Fano cone singularity is at most the dimension of the variety. This is based on its relation with dimensions of moduli spaces of orbifold rational curves. We also propose a conjectural characterization of weighted projective spaces as Fano orbifolds in terms of orbifold rational curves, which would imply that the equality holds only for smooth points.

Keywords

Minimal log discrepancy / Fano cone singularity / orbifold rational curves

Acknowledgments

The first auhtor is partially supported by NSF (Grant No. DMS-2305296). The second auhtor is supported by National Key R&D Program of China under (Grant Nos. 2023YFA-1010500, NSFC-12288201, and NSFC-12231010). The first auhtor thanks Qile Chen for helpful discussions on twisted stable maps and thanks ChrisWoodward for bringing the reference [17] to his attention.

References

[1] Abramovich D, Graber T, Vistoli A. Gromov-witten theory of deligne-mumford stacks. Amer. J. Math., 2008, 130: 1337-1398.

[2] Adem A, Leida J, Ruan Y. Orbifolds and Stringy Topology,vol. 171 of Camb. Tracts Math. Cambridge: Cambridge University Press, 2007.

[3] Ambro F. The minimal log discrepancy. arXiv preprint arXiv: 0611859, 2006.

[4] Abramovich D, Vistoli A. Compactifying the space of stable maps. J. Amer. Math. Soc., 2002, 15(1): 27-75.

[5] Chen J. Characterizing projective spaces for varieties with at most quotient singularities. Bull. Inst. Math. Acad. Sin. (N.S.), 2017, 12(4): 297-314.

[6] Cho K, Miyaoka Y, Shepherd-Barron N I. Characterizations of projective space and applications to complex symplectic manifolds. In Higher Dimensional Birational Geom-etry (Kyoto, 1997), Adv. Stud. Pure Math., 2002, 25: 1-88.

[7] Chen W, Ruan Y.Orbifold gromov-witten theory. In Orbifolds in Mathematics and Physics, Vol. 310 of Contemp. Math., Amer. Math. Soc., Providence, 2002: 25-85.

[8] Chen J, Tseng H H. Cone theorem via Deligne-Mumford stacks. Math. Ann., 2009, 345(3): 525-545.

[9] Debarre O. Springer-Verlag,Higher-Dimensional Algebraic Geometry. New York, 2001.

[10] Eliashberg Y, Givental A, Hofer H. Introduction to symplectic field theory. Modern Birkhser Classics, 2000(Part II): 560-673.

[11] Kebekus S. Characterizing the projective space after cho, miyaoka and shepherd-barron. Complex Geometry, 2002: 147-155.

[12] Kollar J, Mori S. Cambridge Univ. Press,Birational Geometry of Algebraic Varieties. Cambridge, 1998.

[13] Lupercio E, Uribe B. Gerbes over orbifolds and twisted k-theory. Commun. Math., 2004, 245: 449-489.

[14] Li C, Zhou Z. Kähler compactification of Cn and reeb dynamics. Inventiones mathemati-cae., 2026, 243: 181-208.

[15] McLean M. Reeb orbits and the minimal discrepancy of an isolated singularity. Invent. Math., 2016, 204(2): 505-594.

[16] Mori S. Projective manifolds with ample tangent bundles. Ann. Math., 1979, 110: 593-606.

[17] Martens J, Thaddeus M. Variations on a theme of grothendieck. arXiv preprint arXiv: 1210.8161, 2012.

[18] Olsson M. Hom-stacks and restriction scalars. Duke Math. J., 2006, 134(1): 139-164.