## Kyoto Journal of Mathematics

### Nef cone of flag bundles over a curve

#### Abstract

Let $X$ be a smooth projective curve defined over an algebraically closed field $k$, and let $E$ be a vector bundle on $X$. Let ${\mathcal{O}}_{\operatorname {Gr}_{r}(E)}(1)$ be the tautological line bundle over the Grassmann bundle $\operatorname {Gr}_{r}(E)$ parameterizing all the $r$-dimensional quotients of the fibers of $E$. We give necessary and sufficient conditions for ${\mathcal{O}}_{\operatorname {Gr}_{r}(E)}(1)$ to be ample and nef, respectively. As an application, we compute the nef cone of $\operatorname {Gr}_{r}(E)$. This yields a description of the nef cone of any flag bundle over $X$ associated to $E$.

#### Article information

Source
Kyoto J. Math., Volume 54, Number 2 (2014), 353-366.

Dates
First available in Project Euclid: 2 June 2014

https://projecteuclid.org/euclid.kjm/1401741282

Digital Object Identifier
doi:10.1215/21562261-2642422

Mathematical Reviews number (MathSciNet)
MR3215571

Zentralblatt MATH identifier
1302.14025

#### Citation

Biswas, Indranil; Parameswaran, A. J. Nef cone of flag bundles over a curve. Kyoto J. Math. 54 (2014), no. 2, 353--366. doi:10.1215/21562261-2642422. https://projecteuclid.org/euclid.kjm/1401741282

#### References

• [BB] I. Biswas and U. Bruzzo, On semistable principal bundles over a complex projective manifold, Int. Math. Res. Not. IMRN 2008, no. 12, art. ID rnn035.
• [BH] I. Biswas and Y. I. Holla, Semistability and numerically effectiveness in positive characteristic, Internat. J. Math. 22 (2011), 25–46.
• [BP] I. Biswas and A. J. Parameswaran, A criterion for virtual global generation, Ann. Sc. Norm. Super. Pisa Cl. Sci. (5) 5 (2006), 39–53.
• [Br] M. Brion, “The cone of effective one-cycles of certain $G$-varieties” in A Tribute to C. S. Seshadri (Chennai, 2002), Trends Math., Birkhäuser, Basel, 2003, 180–198.
• [DPS] J.-P. Demailly, T. Peternell, and M. Schneider, Compact complex manifolds with numerically effective tangent bundles, J. Algebraic Geom. 3 (1994), 295–345.
• [Fu] T. Fujita, Semipositive line bundles, J. Fac. Sci. Univ. Tokyo Sect. IA Math. 30 (1983), 353–378.
• [Ha] R. Hartshorne, Ample vector bundles, Inst. Hautes Études Sci. Publ. Math. 29 (1966), 63–94.
• [Lan] A. Langer, Semistable sheaves in positive characteristic, Ann. of Math. (2) 159 (2004), 251–276.
• [Laz] R. Lazarsfeld, Positivity in Algebraic Geometry. II. Positivity for Vector Bundles, and Multiplier Ideals, Ergeb. Math. Grenzgeb. (3) 49, Springer, Berlin, 2004.
• [Mi] Y. Miyaoka, “The Chern classes and Kodaira dimension of a minimal variety” in Algebraic Geometry, Sendai, 1985, Adv. Stud. Pure Math. 10, North-Holland, Amsterdam, 1987, 449–476.
• [PS] A. J. Parameswaran and S. Subramanian, “On the spectrum of asymptotic slopes” in Teichmüller Theory and Moduli Problem, Ramanujan Math. Soc. Lect. Notes Ser. 10, Ramanujan Math. Soc., Mysore, India, 2010, 519–528.
• [RR] S. Ramanan and A. Ramanathan, Some remarks on the instability flag, Tohoku Math. J. (2) 36 (1984), 269–291.