Higher Fano manifolds and rational surfaces. (English) Zbl 1124.14039

If \(X\) is a Fano manifold, the pseudoindex of \(X\) is the minimal \(c_1(T_X)\) degree of a rational curve on \(X\). The Fano manifold \(X\) is called 2-Fano if the 2-nd graded piece of the Chern character \(\text{ch}_2(X) = 1/2(c_1(T_X)^2-2c_2(T_X))\) is nef. Examples of 2-Fano 3-folds are studied in more detail in the authors’ e-print [A note on Fano manifolds whose second Chern character is positive; arXiv:math.AG/0602644]. In particular, a smooth complete intersection \(X \subset {\mathbb P}^n\) of hypersurfaces of degrees \(d_1,\dots,d_s\) is Fano iff \(d_1+\cdots +d_s \leq n\), and \(X\) is 2-Fano iff \(d_1^2 + \cdots +d_s^2 \leq n\).
By Theorem 1.2, for a 2-Fano manifold \(X\) the irreducibility of some spaces of rational curves on \(X\) implies that the general point of \(X\) is contained in a rational surface. The proof uses the bend-and-break approach of Y. Miyaoka and S. Mori [Ann. Math. (2) 124, 65–69 (1986; Zbl 0606.14030)]. In Section 3 the authors prove that the conditions of Theorem 1.2 are fulfilled for certain Fano manifolds of pseudoindex 3. The main application of these results concerns the 2-Fano complete intersections in projective spaces. By Proposition 1.4, if \(X\) is any smooth 2-Fano complete intersection in \({\mathbb P}^n\) then the general point \(x \in X\) lies in a rational surface. Moreover, every irreducible component of the moduli space \(\bar{\mathcal M}_{o,o}(X)\) of genus 0 stable maps to \(X\), whose general point parametrizes a free curve [see C. Araujo and J. Kollár, Bolyai Soc. Math. Stud. 12, 13–68 (2003; Zbl 1080.14521); arXiv:math.AG/0203173], is uniruled.
In Section 4 are given 2 examples which show that the conditions of the results in the paper are sharp. The first example is the cubic fourfold \(X \subset {\mathbb P}^4\), which is Fano but not 2-Fano. By a recent result of the authors, for the general cubic 4-fold \(X\) any component of \(\bar{\mathcal M}_{o,o}(X)\) parametrizes a free curve, but infinitely many of these components are not uniruled [A. J. de Jong and J. Starr, Ill. J. Math. 48, No. 2, 415–450 (2004; Zbl 1081.14007)]. This shows that in Proposition 1.4 one can’t replace “2-Fano” by “Fano”.


14J45 Fano varieties
14H10 Families, moduli of curves (algebraic)
14J10 Families, moduli, classification: algebraic theory
Full Text: DOI arXiv


[1] F. Campana, Connexité rationnelle des variétés de Fano , Ann. Sci. École Norm. Sup. (4) 25 (1992), 539–545. · Zbl 0783.14022
[2] I. Coskun, J. Harris, and J. Starr, The ample cone of the Kontsevich moduli space , to appear in Canad. J. Math., preprint, 2005.
[3] A. J. De Jong and J. M. Starr, Cubic fourfolds and spaces of rational curves , Illinois J. Math. 48 (2004), 415–450. · Zbl 1081.14007
[4] -, A note on Fano manifolds whose second Chern character is positive , preprint,\arxivmath/0602644v1[math.AG]
[5] W. Fulton and R. Pandharipande, “Notes on stable maps and quantum cohomology” in Algebraic Geometry (Santa Cruz, Calif., 1995) , Proc. Sympos. Pure Math. 62 , Part 2, Amer. Math. Soc., Providence, 1997, 45–96. · Zbl 0898.14018
[6] T. Graber, J. Harris, and J. M. Starr, Families of rationally connected varieties , J. Amer. Math. Soc. 16 (2003), 57–67. JSTOR: · Zbl 1092.14063
[7] J. Harris, M. Roth, and J. M. Starr, Rational curves on hypersurfaces of low degree , J. Reine Angew. Math. 571 (2004), 73–106. · Zbl 1052.14027
[8] B. Kim and R. Pandharipande, “The connectedness of the moduli space of maps to homogeneous spaces” in Symplectic Geometry and Mirror Symmetry (Seoul, 2000) , World Sci., River Edge, N.J., 2001, 187–201. · Zbl 1076.14517
[9] J. KolláR, Rational Curves on Algebraic Varieties , Ergeb. Math. Grenzgeb. (3) 32 , Springer, Berlin, 1996.
[10] J. KolláR, Y. Miyaoka, and S. Mori, Rational connectedness and boundedness of Fano manifolds , J. Differential Geom. 36 (1992), 765–779. · Zbl 0759.14032
[11] M. Kontsevich, “Enumeration of rational curves via torus actions” in The Moduli Space of Curves (Texel Island, Netherlands, 1994) , Progr. Math. 129 , Birkhäuser, Boston, 1995, 335–368. · Zbl 0885.14028
[12] Y. Miyaoka and S. Mori, A numerical criterion for uniruledness , Ann. of Math. (2) 124 (1986), 65–69. JSTOR: · Zbl 0606.14030
[13] A. M. Mustaţǎ and A. Mustaţǎ, Universal relations on stable map spaces in genus zero , preprint,\arxivmath/0607431v2[math.AG]
[14] A. Parker, An elementary GIT construction of the moduli space of stable maps , preprint,\arxivmath/0604092v1[math.AG] · Zbl 1397.83016
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.