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Let $X \subset \mathbb{P}^n$ be a non-singular complete intersection of $s$ hypersurfaces of degrees $d_1,\dots,d_s$ over an algebraically closed field $k$ of characteristic zero. Let $d=d_1 + \dots + d_s$. Suppose that $X$ is Fano, i.e. $d < n + 1$. Denote by $F_1(X)$ the Fano variety of lines in $X$. Then it is known that if $X$ is general, then one has

$$\dim F_1(X) = 2n - d - s - 2 \quad (*),$$

if this number is non-negative. Moreover, a conjecture of Debarre and de Jong states that $(*)$ holds for all such $X$ (not necessarily general) if $s=1$, i.e. if $X$ is a hypersurface. I have only seen this conjecture stated for hypersurfaces. My question is what happens for complete intersections.

Is $(*)$ still expected to hold for all such $X$ (not necessarily general) if $s >1$?

Perhaps this is dealt with in the original article where Debarre and de Jong made this conjecture, however I have been unable to find it, only other papers (not by Debarre and de Jong) which state this conjecture (without reference).

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  • $\begingroup$ Your formula (*) is wrong. It should be $2n-d-s-2$. $\endgroup$ Nov 7, 2013 at 16:32
  • $\begingroup$ @Jason: You are quite right, thank you. $\endgroup$ Nov 7, 2013 at 16:35
  • $\begingroup$ I guess the first test case is whether a complete intersection of $s=2t$ quadrics in $\mathbb{P}^{4t}$ can contain a 1-parameter family of linear $\mathbb{P}^t$s. For hypersurfaces, independently, Debarre and I proved this cannot happen (my argument is in an appendix to an article of Browning and Heath-Brown). Perhaps the same argument would apply in this case. At any rate, I am unaware of anybody investigating this. It sounds like a great project! $\endgroup$ Nov 7, 2013 at 16:44
  • $\begingroup$ Thanks for your comments Jason. I won't deny when I put this on mathoverflow I was hoping you would see it... Is there an original paper by Debarre and/or de Jong where they state this conjecture, or is it more of a word-of-mouth conjecture? $\endgroup$ Nov 7, 2013 at 16:57
  • $\begingroup$ Actually I recommend checking with Roya Beheshti at Washington University. She knows much more about this than I do. $\endgroup$ Nov 7, 2013 at 18:17

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