Comptes Rendus
Algebraic geometry
Compactifications of conic spaces in del Pezzo 3-fold
Comptes Rendus. Mathématique, Volume 357 (2019) no. 9, pp. 729-736.

Let V5 be the del Pezzo 3-fold defined by the 6-dimensional linear section of the Grassmannian variety Gr(2,5) under the Plücker embedding. In this paper, we present an explicit birational relation of compactifications of degree-two rational curves (i.e. conics) in V5. By a product, we obtain the virtual Poincaré polynomial of compactified moduli spaces.

Soit V5 le del Pezzo 3 défini par la section linéaire de dimension 6 de la variété grassmannienne Gr(2,5) située sous l'enrobage de Plücker. Dans cet article, nous présentons une relation birationnelle explicite de compactifications de courbes rationnelles de degré deux en V5. Au moyen d'un produit, nous obtenons le polynôme de Poincaré virtuel des espaces de modules compactifiés.

Published online:
DOI: 10.1016/j.crma.2019.09.003

Kiryong Chung 1; Sang-Bum Yoo 2

1 Department of Mathematics Education, Kyungpook National University, 80 Daehakro, Bukgu, Daegu 41566, Republic of Korea
2 School of Natural Science, UNIST, 50 UNIST-gil, Ulsan 44919, Republic of Korea
     author = {Kiryong Chung and Sang-Bum Yoo},
     title = {Compactifications of conic spaces in del {Pezzo} 3-fold},
     journal = {Comptes Rendus. Math\'ematique},
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Kiryong Chung; Sang-Bum Yoo. Compactifications of conic spaces in del Pezzo 3-fold. Comptes Rendus. Mathématique, Volume 357 (2019) no. 9, pp. 729-736. doi : 10.1016/j.crma.2019.09.003.

[1] V. Alexeev; A. Knutson Complete moduli spaces of branchvarieties, J. Reine Angew. Math., Volume 639 (2010), pp. 39-71

[2] B. Bakker; A. Jorza Higher rank stable pairs on k3 surfaces, Commun. Number Theory Phys., Volume 6 (2012) no. 4, pp. 805-847

[3] K. Chung A desingularization of Kontsevich's compactification of twisted cubics in V5, 2019 | arXiv

[4] K. Chung; J. Hong; Y.-H. Kiem Compactified moduli spaces of rational curves in projective homogeneous varieties, J. Math. Soc. Jpn., Volume 64 (2012) no. 4, pp. 1211-1248

[5] K. Chung; J. Hong; S. Lee Geometry of moduli spaces of rational curves in linear sections of Grassmannian Gr(2,5), J. Pure Appl. Algebra, Volume 222 (2018) no. 4, pp. 868-888

[6] K. Chung; Y.-H. Kiem Hilbert scheme of rational cubic curves via stable maps, Amer. J. Math., Volume 133 (2011) no. 3, pp. 797-834

[7] K. Chung; H.-B. Moon Mori's program for the moduli space of conics in Grassmannian, Taiwan. J. Math., Volume 21 (June 2017) no. 3, pp. 621-652

[8] D. Faenzi Bundles over the Fano threefold V5, Commun. Algebra, Volume 33 (2005) no. 9, pp. 3061-3080

[9] W. Fulton; R. Pandharipande Notes on stable maps and quantum cohomology, Algebraic Geometry—Santa Cruz 1995, Proc. Sympos. Pure Math., vol. 62, Amer. Math. Soc., Providence, RI, USA, 1997, pp. 45-96

[10] M. Furushima; N. Nakayama The family of lines on the Fano threefold V5, Nagoya Math. J., Volume 116 (1989), pp. 111-122

[11] D.R. Grayson; M.E. Stillman Macaulay2, a software system for research in algebraic geometry (available at)

[12] R. Hartshorne Stable reflexive sheaves, Math. Ann., Volume 254 (1980), pp. 121-176

[13] D. Huybrechts; M. Lehn The Geometry of Moduli Spaces of Sheaves, Cambridge Mathematical Library, Cambridge University Press, Cambridge, UK, 2010

[14] A. Iliev The Fano surface of the Gushel threefold, Compos. Math., Volume 94 (1994) no. 1, pp. 81-107

[15] Y.-H. Kiem Hecke correspondence, stable maps, and the Kirwan desingularization, Duke Math. J., Volume 136 (2007) no. 3, pp. 585-618

[16] B. Kim; R. Pandharipande The connectedness of the moduli space of maps to homogeneous spaces, Seoul, 2000, World Sci. Publ., River Edge, NJ, USA (2001), pp. 187-201

[17] J. Li; G. Tian Virtual moduli cycles and Gromov–Witten invariants of algebraic varieties, J. Amer. Math. Soc., Volume 11 (1998) no. 1, pp. 119-174

[18] H.-B. Moon Birational Geometry of Moduli Spaces of Curves of Genus Zero, Seoul National University, 2011 (PhD thesis)

[19] V. Muñoz Hodge polynomials of the moduli spaces of rank 3 pairs, Geom. Dedic., Volume 136 (2008), pp. 17-46

[20] A.E. Parker An elementary GIT construction of the moduli space of stable maps, Ill. J. Math., Volume 51 (2007) no. 3, pp. 1003-1025

[21] G. Sanna Rational Curves and Instantons on the Fano Threefold Y5, 2014 (PhD thesis)

[22] M. Thaddeus Geometric invariant theory and flips, J. Amer. Math. Soc., Volume 9 (1996) no. 3, pp. 691-723

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