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Let $X_d\subset \mathbb P^{n+1}_{\mathbb C}$ be a smooth hypersurface of ddegree $d$. Harris, Mazur and Pandharipande proved that there is a bound $b(d)$ such that if $n>b(d)$, $X_d$ is unirational. Is there a formula for the degree (in function of $d$) of the unirational parametrization one gets with their method?

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  • $\begingroup$ I do not remember the bound, but I remember that it is an iterated exponential in $d$. You might also look at the article by Paranjape and Srinivas, who prove roughly the same result (the basic technique goes back to Morin and Predonzan). $\endgroup$ Jul 20, 2017 at 11:41

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