Rational curves on hypersurfaces
-- Yuan Wang, April 20, 2018

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It is a well known fact that a general hyperplane of degree d in P^n
is rationally connected if d <= n, but contains very few curves if d >= n + 1. 
More generally, let X be a smooth projective variety and H be a hypersurface of X 
such that K_X + H is anti-ample, then by the adjunction formula and a classical result of 
Koll&aacute;r-Miyaoka-Mori we know that H is rationally connected. In a recent project we use the minimal model program as well as other techniques in birational geometry to study further how the behavior of rational curves on X as well as the positivity of -(K_X + H) and H influence the behavior of rational curves on H. In this talk I will present several results and examples of this kind. In particular we will see criteria for uniruledness and rational connectedness of H.