Rational Curves on Blowing-ups of Projective Spaces

  • Published 2006


Let Morβ(P, Y ) denote the moduli space of morphisms f from a complex projective line P1 to a smooth complex projective variety Y such that f∗[P] = β where β is a given second homology class of Y . We study the irreducibility and the rational connectedness of the moduli space when Y is a successive blowing-up of a product of projective spaces with a suitable condition on β. To state the Main Theorem proven in this paper, let us introduce some notation. LetX = ∏m k=1 P nk , X0 = X and let πi : Xi → Xi−1, i = 1, ..., r, be a blowing-up of Xi−1 along a smooth irreducible subvariety Zi. Let E t i ⊂ Xr be the total transform (πi ◦ ...◦πr) Zi of the exceptional divisor associated to Zi and let Hk be the divisor class coming from the hyperplane class of the k-th projective space Pnk . Letmi = #{Zj | j < i, (πj◦...◦πr)(Zj) ⊃ Et i}. So general points of Zi are the mi-th infinitesimal points of X. Denote by Morβ(P,Xr) the open sublocus of Morβ(P,Xr), consisting of f whose image does not lie on exceptional divisors: f(P1) * Et i , ∀ i.

Cite this paper

@inproceedings{KIM2006RationalCO, title={Rational Curves on Blowing-ups of Projective Spaces}, author={BUMSIG KIM}, year={2006} }