×

The cone of curves of Fano varieties of coindex four. (English) Zbl 1111.14037

A (complex) Fano manifold is a smooth complex projective variety \(X\) with ample anticanonical class \(-K_X\). The index \(r_X\) of \(X\) is the largest integer \(m\) such that \(-K_X = mH\) for some divisor \(H\) on \(X\), and the coindex of \(X\) is defined by \(\dim X + 1 - r_X\). By a known result of S. Kobayashi and T. Ochiai [J. Math. Kyoto Univ. 13, 31–47 (1973; Zbl 0261.32013)], the coindex of \(X\) is always non-negative, and the only Fano varieties of conindex \(0\) and \(1\) are respectively the projective spaces and the quadrics. The Fano manifolds of coindex 2 are classified by T. Fujita [J. Math. Soc. Japan 33, 415–434 (1981; Zbl 0474.14018)], those of coindex 3 are classified by S. Mukai [Proc. Natl. Acad. Sci. USA 86, No. 9, 3000–3002 (1989; Zbl 0679.14020)] and M. Mella [J. Algebr. Geom. 8, No. 2, 197–206 (1999; Zbl 0970.14023)], while the classification of Fano manifolds of any coindex \(\geq 4\) is unknown.
In this paper is studied the structure of the cone of rational curves on Fano manifolds \(X\) of dimension \(\geq 5\) and coindex \(4\) in the case when the Picard number \(\rho_X \geq 2\). By Theorem 1.1, the main result of the paper, any such \(X\) has dimension \(n = \dim X \leq 8\), the cone NE\((X)\) of effective 1-cycles on \(X\) is generated by \(\rho_X\) extremal rays, and \(\text{NE}(X)\) can belong to 23 possible types, depending on \(n = \dim X\), the Picard number \(\rho_X\), and the types of extremal contractions generating \(NE(X)\). The examples given throughout the paper show that there exist varieties that belong to any of these types.

MSC:

14J45 Fano varieties
14E30 Minimal model program (Mori theory, extremal rays)
PDFBibTeX XMLCite
Full Text: DOI arXiv

References:

[1] DOI: 10.2478/BF02476544 · Zbl 1068.14049 · doi:10.2478/BF02476544
[2] Andreatta M., Nagoya Math. J. 168 pp 127– · Zbl 1055.14015 · doi:10.1017/S0027763000008400
[3] DOI: 10.4310/AJM.2005.v9.n4.a5 · Zbl 1100.14033 · doi:10.4310/AJM.2005.v9.n4.a5
[4] Andreatta M., J. Algebraic Geom. 7 pp 253–
[5] Campana F., Ann. Sci. École Norm. Sup. 25 pp 539– · Zbl 0783.14022 · doi:10.24033/asens.1658
[6] DOI: 10.1007/978-1-4757-5406-3 · doi:10.1007/978-1-4757-5406-3
[7] DOI: 10.2307/2374881 · Zbl 0708.14009 · doi:10.2307/2374881
[8] DOI: 10.2969/jmsj/03330415 · Zbl 0474.14018 · doi:10.2969/jmsj/03330415
[9] DOI: 10.1017/CBO9780511662638 · doi:10.1017/CBO9780511662638
[10] DOI: 10.1007/978-3-642-56202-0_10 · doi:10.1007/978-3-642-56202-0_10
[11] Kobayashi S., J. Math. Kyoto Univ. 13 pp 31–
[12] DOI: 10.1007/978-3-662-03276-3 · doi:10.1007/978-3-662-03276-3
[13] Kollár J., J. Differential. Geom. 36 pp 765– · Zbl 0759.14032 · doi:10.4310/jdg/1214453188
[14] Mella M., J. Algebraic Geom. 8 pp 197–
[15] S. Mukai, Birational Geometry of Algebraic Varieties (1988) pp. 60–67.
[16] DOI: 10.1073/pnas.86.9.3000 · Zbl 0679.14020 · doi:10.1073/pnas.86.9.3000
[17] DOI: 10.2977/prims/1195193917 · Zbl 0234.32017 · doi:10.2977/prims/1195193917
[18] DOI: 10.4153/CMB-2006-028-3 · Zbl 1115.14034 · doi:10.4153/CMB-2006-028-3
[19] Szurek M., Nagoya Math. J. 120 pp 89– · Zbl 0728.14037 · doi:10.1017/S0027763000003275
[20] Wiśniewski J. A., Bull. Polish Acad. Sci. Math. 38 pp 173–
[21] DOI: 10.1007/BF02568756 · Zbl 0715.14033 · doi:10.1007/BF02568756
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.