JournalscmhVol. 86, No. 3pp. 593–607

Rational connectedness modulo the non-nef locus

  • Amaël Broustet

    Université Lille I, Villeneuve d'Ascq, France
  • Gianluca Pacienza

    Université de Strasbourg, France
Rational connectedness modulo the non-nef locus cover
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Abstract

It is well known that a smooth projective Fano variety is rationally connected. Recently Zhang [Z2] (and later Hacon and McKernan [HM] as a special case of their work on the Shokurov RC-conjecture) proved that the same conclusion holds for a klt pair (X,Δ)(X,\Delta) such that (KX+Δ)-(K_X+\Delta) is big and nef. We prove here a natural generalization of the above result by dropping the nefness assumption. Namely we show that a klt pair (X,Δ)(X,\Delta) such that (KX+Δ)-(K_X+\Delta) is big is rationally connected modulo the non-nef locus of (KX+Δ)-(K_X+\Delta). This result is a consequence of a more general structure theorem for arbitrary pairs (X,Δ)(X,\Delta) with (KX+Δ)-(K_X+\Delta) pseff.

Cite this article

Amaël Broustet, Gianluca Pacienza, Rational connectedness modulo the non-nef locus. Comment. Math. Helv. 86 (2011), no. 3, pp. 593–607

DOI 10.4171/CMH/235