Taiwanese Journal of Mathematics

Mori's Program for the Moduli Space of Conics in Grassmannian

Kiryong Chung and Han-Bom Moon

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We complete Mori's program for Kontsevich's moduli space of degree $2$ stable maps to the Grassmannian of lines. We describe all birational models in terms of moduli spaces (of curves and sheaves), incidence varieties, and Kirwan's partial desingularization.

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Taiwanese J. Math., Volume 21, Number 3 (2017), 621-652.

First available in Project Euclid: 1 July 2017

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Primary: 14D22: Fine and coarse moduli spaces 14F42: Motivic cohomology; motivic homotopy theory [See also 19E15] 14E15: Global theory and resolution of singularities [See also 14B05, 32S20, 32S45]

moduli space rational curves Grassmannian birational geometry


Chung, Kiryong; Moon, Han-Bom. Mori's Program for the Moduli Space of Conics in Grassmannian. Taiwanese J. Math. 21 (2017), no. 3, 621--652. doi:10.11650/tjm/7769. https://projecteuclid.org/euclid.twjm/1498874610

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  • A. Bertram, C. Martinez and J. Wang, The birational geometry of moduli spaces of sheaves on the projective plane, Geom. Dedicata 173 (2014), no. 1, 37–64.
  • D. Chen, Mori's program for the Kontsevich moduli space $\ol{\ms{M}}_{0,0}(\mb{P}^3,3)$, Int. Math. Res. Not. IMRN 2008, Art. ID rnn 067, 17 pp.
  • D. Chen and I. Coskun, Stable base locus decompositions of Kontsevich moduli spaces, Michigan Math. J. 59 (2010), no. 2, 435–466.
  • ––––, Towards Mori's program for the moduli space of stable maps, Amer. J. Math. 133 (2011), no. 5, 1389–1419.
  • J. Choi and K. Chung, The geometry of the moduli space of one-dimensional sheaves, Sci. China Math. 58 (2015), no. 3, 487–500.
  • ––––, Moduli spaces of $\alpha$-stable pairs and wall-crossing on $\mb{P}^2$, J. Math. Soc. Japan 68 (2016), no. 2, 685–709.
  • J. Choi and M. Maican, Torus action on the moduli spaces of torsion plane sheaves of multiplicity four, J. Geom. Phys. 83 (2014), 18–35.
  • K. Chung, J. Hong and Y.-H. Kiem, Compactified moduli spaces of rational curves in projective homogeneous varieties, J. Math. Soc. Japan 64 (2012), no. 4, 1211–1248.
  • K. Chung and H.-B. Moon, Moduli of sheaves, Fourier-Mukai transform, and partial desingularization, Math. Z. 283 (2016), no. 1-2, 275–299.
  • I. Coskun and J. Starr, Divisors on the space of maps to Grassmannians, Int. Math. Res. Not. 2006, Art. ID 35273, 25 pp.
  • A. J. de Jong and J. Starr, Cubic fourfolds and spaces of rational curves, Illinois J. Math. 48 (2004), no. 2, 415–450.
  • J.-M. Drézet, Fibrés exceptionnels et variétés de modules de faisceaux semi-stables sur $\mb{P}_2(\mb{C})$, J. Reine Angew. Math. 380 (1987), 14–58.
  • J.-M. Drézet and M. Maican, On the geometry of the moduli spaces of semi-stable sheaves supported on plane quartics, Geom. Dedicata 152 (2011), no. 1, 17–49.
  • A. B. Givental, Equivariant Gromov-Witten invariants, Internat. Math. Res. Notices 1996, no. 13, 613–663.
  • S. Hosono and H. Takagi, Double quintic symmetroids, Reye congruences, and their derived equivalence, J. Differential Geom. 104 (2016), no. 3, 443–497.
  • ––––, Geometry of symmetric determinantal loci, arXiv:1508.01995, 2015.
  • O. Iena, A global description of the fine Simpson moduli space of $1$-dimensional sheaves supported on plane quartics, arXiv:1607.01319, 2016.
  • A. Iliev and L. Manivel, Fano manifolds of degree ten and EPW sextics, Ann. Sci. Éc. Norm. Supér. (4) 44 (2011), no. 3, 393–426.
  • Y.-H. Kiem, Hecke correspondence, stable maps, and the Kirwan desingularization, Duke Math. J. 136 (2007), no. 3, 585–618.
  • Y.-H. Kiem and H.-B. Moon, Moduli space of stable maps to projective space via GIT, Internat. J. Math. 21 (2010), no. 5, 639–664.
  • F. C. Kirwan, Partial desingularisations of quotients of nonsingular varieties and their Betti numbers, Ann. of Math. (2) 122 (1985), no. 1, 41–85.
  • A. Kuznetsov and D. Markushevich, Symplectic structures on moduli spaces of sheaves via the Atiyah class, J. Geom. Phys. 59 (2009), no. 7, 843–860.
  • J. Le Potier, Systèmes cohérents et structures de niveau, Astérisque 214 (1993), 143 pp.
  • A. López Martín, Poincaré polynomials of stable map spaces to Grassmannians, Rend. Semin. Mat. Univ. Padova 131 (2014), 193–208.
  • H.-B. Moon, Mori's program for $\ol{M}_{0,7}$ with symmetric divisors, to appear in Canad. J. Math., 35 pp.
  • H.-B. Moon and S.-B. Yoo, Birational geometry of the moduli space of rank $2$ parabolic vector bundles on a rational curve, Int. Math. Res. Not. IMRN 2016 (2016), no. 3, 827–859.
  • V. Muñoz, Hodge polynomials of the moduli spaces of rank $3$ pairs, Geom. Dedicata 136 (2008), no. 1, 17–46.
  • K. G. O'Grady, Irreducible symplectic $4$-folds and Eisenbud-Popescu-Walter sextics, Duke Math. J. 134 (2006), no. 1, 99–137.
  • D. Oprea, Divisors on the moduli spaces of stable maps to flag varieties and reconstruction, J. Reine Angew. Math. 586 (2005), 169–205.
  • A. E. Parker, An elementary GIT construction of the moduli space of stable maps, Illinois J. Math. 51 (2007), no. 3, 1003–1025.
  • Y. G. Prokhorov, Compactifications of $\mb{C}^4$ of index $3$, in Algebraic Geometry and its Applications, (Yaroslavl', 1992), 159–169, Aspects Math. E25, Vieweg, Braunschweig, 1994.
  • J. Stoppa, Universal covers and the GW/Kronecker correspondence, Commun. Number Theory Phys. 5 (2011), no. 2, 353–395.
  • Y. Yuan, Moduli spaces of semistable sheaves of dimension $1$ on $\mb{P}^2$, Pure Appl. Math. Q. 10 (2014), no. 4, 723–766.