I would like to know what are the open big and interesting problems related to moduli spaces and moduli stacks ?
Thanks in advance for your help.
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$\begingroup$ I have added a "big-list" tag, I hope you don't mind. $\endgroup$– M.G.Commented Aug 9, 2018 at 20:22
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2$\begingroup$ Is $M_{g}$ of general type for $g<22$? $\endgroup$– Ennio Mori coneCommented Aug 9, 2018 at 22:21
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1 Answer
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The Debarre-de Jong conjecture: if $\mathrm{X}\subset\mathbf{P}^n$ is a smooth hypersurface of degree $d\leqslant n$, then the dimension of the moduli space of lines on $\mathrm{X}$ is the expected one, namely $2n-d-3$.
It is known to be true for at least $d\leqslant 6$.
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4$\begingroup$ It’s known to be true for $d\leq 8$ by work of Beheshti. $\endgroup$ Commented Aug 9, 2018 at 22:00
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$\begingroup$ @SamirCanning you are right, I have seen Beheshti's paper before, but had the wrong number in mind. Apparently 8 is still the state-of-the-art. $\endgroup$– ssxCommented Aug 10, 2018 at 12:34
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$\begingroup$ Another comment about this conjecture that confused me at first was that, like we did, people always phrase what’s known about the conjecture in terms of the degree of the hypersurface instead of the dimension of the projective space. I always thought that was odd because to me it seemed like at first glance that it would be harder to prove the conjecture for a sextic in $\mathbb{P}^{1000}$ than a degree 9 in $\mathbb{P}^9$. But actually the conjecture reduces to proving the $n=d$ case by intersecting with a $d$-plane and a short argument. $\endgroup$ Commented Aug 10, 2018 at 15:23