# Varieties dominated by products of curves

Let $X$ be an irreducible smooth projective variety of dimension $d$. Do there exist irreducible smooth projective curves $C_1, C_2,\ldots, C_d$, an open subset $U\subset C_1\times C_2\times\ldots\times C_d$ and a dominant morphism $f:U\to X$.

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This is not true. For example, this does not hold for sufficiently general hypersurfaces of large degree (and dimension $> 1$) by results of C. Schoen "Varieties dominated by product varieties." Internat. J. Math. 7 (1996), no. 4, 541–571. –  ulrich Jun 4 '12 at 14:30
Tony Scholl's comment at mathoverflow.net/questions/33665/… looks relevant. –  David Speyer Jun 4 '12 at 18:51