Bulletin of the Korean Mathematical Society (대한수학회보)

Korean Mathematical Society (대한수학회)
권호 표지
DOI QR코드

DOI QR Code

IRREDUCIBILITY OF THE MODULI SPACE FOR THE QUOTIENT SINGULARITY $\frac{1}{2k+1}(k+1,1,2k)$

  • Seung-Jo, Jung (Department of Mathematics Education, and Institute of Pure and Applied Mathematics Jeonbuk National University)
  • Received : 2021.11.02
  • Accepted : 2022.07.15
  • Published : 2022.11.30
Download PDF
⟨ Previous Next ⟩

Abstract

A 3-fold quotient terminal singularity is of the type $\frac{1}{r}(b,1,-1)$ with gcd(r, b) = 1. In [6], it is proved that the economic resolution of a 3-fold terminal quotient singularity is isomorphic to a distinguished component of a moduli space 𝓜𝜃 of 𝜃-stable G-constellations for a suitable 𝜃. This paper proves that each connected component of the moduli space 𝓜𝜃 has a torus fixed point and classifies all torus fixed points on 𝓜𝜃. By product, we show that for $\frac{1}{2k+1}(k+1,1,-1)$ case the moduli space 𝓜𝜃 is irreducible.

Keywords

Acknowledgement

This work was partially supported by NRF grant (NRF-2021R1C1C1004097) of the Korean government.

References

  1. T. Bridgeland, A. King, and M. Reid, The McKay correspondence as an equivalence of derived categories, J. Amer. Math. Soc. 14 (2001), no. 3, 535-554. https://doi.org/10.1090/S0894-0347-01-00368-X
  2. A. Craw and A. Ishii, Flops of G-Hilb and equivalences of derived categories by variation of GIT quotient, Duke Math. J. 124 (2004), no. 2, 259-307. https://doi.org/10.1215/S0012-7094-04-12422-4
  3. A. Craw, D. Maclagan, and R. R. Thomas, Moduli of McKay quiver representations. I. The coherent component, Proc. Lond. Math. Soc. (3) 95 (2007), no. 1, 179-198. https://doi.org/10.1112/plms/pdm009
  4. V. I. Danilov, Birational geometry of three-dimensional toric varieties, Izv. Akad. Nauk SSSR Ser. Mat. 46 (1982), no. 5, 971-982, 1135.
  5. S.-J. Jung, McKay quivers and terminal quotient singularities in dimension 3, PhD thesis, University of Warwick, 2014.
  6. S.-J. Jung, Terminal quotient singularities in dimension three via variation of GIT, J. Algebra 468 (2016), 354-394. https://doi.org/10.1016/j.jalgebra.2016.08.032
  7. O. Kedzierski, Danilov's resolution and representations of the McKay quiver, Tohoku Math. J. (2) 66 (2014), no. 3, 355-375. https://doi.org/10.2748/tmj/1412783203
  8. A. D. King, Moduli of representations of finite-dimensional algebras, Quart. J. Math. Oxford Ser. (2) 45 (1994), no. 180, 515-530. https://doi.org/10.1093/qmath/45.4.515
  9. I. Nakamura, Hilbert schemes of abelian group orbits, J. Algebraic Geom. 10 (2001), no. 4, 757-779.
  10. M. Reid, Young person's guide to canonical singularities, in Algebraic geometry, Bowdoin, 1985 (Brunswick, Maine, 1985), 345-414, Proc. Sympos. Pure Math., 46, Part 1, Amer. Math. Soc., Providence, RI, 1987.