Bulletin of the Korean Mathematical Society (대한수학회보)

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DOUBLE LINES IN THE QUINTIC DEL PEZZO FOURFOLD

  • Kiryong Chung (Department of Mathematics Education Kyungpook National University)
  • Received : 2022.04.05
  • Accepted : 2022.07.15
  • Published : 2023.03.31
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Abstract

Let Y be the quintic del Pezzo 4-fold defined by the linear section of Gr(2, 5) by ℙ7. In this paper, we describe the locus of double lines in the Hilbert scheme of coincs in Y. As a corollary, we obtain the desigularized model of the moduli space of stable maps of degree 2 in Y. We also compute the intersection Poincaré polynomial of the stable map space.

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References

  1. K. Chung, Desingularization of Kontsevich's compactification of twisted cubics in V5, Manuscripta Mathematica, 2022.
  2. 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. https://doi.org/10.2969/jmsj/06441211
  3. K. Chung, J. Hong, and S. Lee, Geometry of moduli spaces of rational curves in linear sections of Grassmannian Gr(2, 5), J. Pure Appl. Algebra 222 (2018), no. 4, 868-888. https://doi.org/10.1016/j.jpaa.2017.05.011
  4. K. Chung and Y.-H. Kiem, Hilbert scheme of rational cubic curves via stable maps, Amer. J. Math. 133 (2011), no. 3, 797-834. https://doi.org/10.1353/ajm.2011.0021
  5. K. Chung and H.-B. Moon, Mori's program for the moduli space of conics in Grassmannian, Taiwanese J. Math. 21 (2017), no. 3, 621-652. https://doi.org/10.11650/tjm/7769
  6. A. Fujiki and S. Nakano, Supplement to "On the inverse of monoidal transformation", Publ. Res. Inst. Math. Sci. 7 (1971/72), 637-644. https://doi.org/10.2977/prims/1195193401
  7. W. Fulton and R. Pandharipande, Notes on stable maps and quantum cohomology, In Algebraic geometry-Santa Cruz 1995, volume 62 of Proc. Sympos. Pure Math., pages 45-96. Amer. Math. Soc., Providence, RI, 1997.
  8. D. Grayson and M. Stillman, Macaulay2, a software system for research in algebraic geometry, Available at http://www.math.uiuc.edu/Macaulay2/.
  9. A. G. Kuznetsov, Y. G. Prokhorov, and C. A. Shramov, Hilbert schemes of lines and conics and automorphism groups of Fano threefolds, Jpn. J. Math. 13 (2018), no. 1, 109-185. https://doi.org/10.1007/s11537-017-1714-6
  10. L. G. Maxim, Intersection homology and perverse sheaves, with applications to singularities, Book Project, 2018.
  11. Y. Prokhorov, Compactifications of ℂ4 of index 3, In Algebraic geometry and its applications (Yaroslavl , 1992) Aspects Math., E25:159-169, Vieweg, Braunschweig, 1994. https://doi.org/10.1007/978-3-322-99342-7_13
  12. Y. Prokhorov and M. Zaidenberg, Examples of cylindrical Fano fourfolds, Eur. J. Math. 2 (2016), no. 1, 262-282. https://doi.org/10.1007/s40879-015-0051-7
  13. V. V. Przyjalkowski, I. A. Cheltsov, and C. A. Shramov, Fano threefolds with infinite automorphism groups, Izv. Math. 83 (2019), no. 4, 860-907; translated from Izv. Ross. Akad. Nauk Ser. Mat. 83 (2019), no. 4, 226-280. https://doi.org/10.4213/im8834