Recall that a $K3$ surface is called exceptional if its Picard number is 20. The Fermat quartic $K3$ surface in $\mathbb P^3$ is exceptional. My question is,
Are there infinitely many non-isomorphic quartic exceptional $K3$ surfaces?
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$\begingroup$ Just to clarify, when you say "non-isomorphic", do you mean non-isomorphic as compact complex surfaces, i.e., not biholomorphic, or do you mean non-isomorphic as quartic hypersurfaces, i.e., not projectively equivalent? Some K3 surfaces have infinitely many non-projectively equivalent embeddings in the same projective space. $\endgroup$– Jason StarrCommented Sep 18, 2023 at 11:32
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$\begingroup$ By "non-isomorphic", I mean non-isomorphic as compact complex surfaces. However, if the answer to the original question in the post is not easy, I also would like to know if there are infinitely many quartic exceptional $K3$ surfaces that are not projectively equivalent. $\endgroup$– BasicsCommented Sep 18, 2023 at 11:48
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$\begingroup$ These are more often called "singular K3 surfaces", and a theorem of Shioda-Inose classifies them as resolutions of Kummer quotients of products of isogenous elliptic curves. Indeed there are infinitely many of them (up to isomorphism) even though they do not deform in moduli. See Section 14.3.4 in Huybrechts' "Lectures on K3 surfaces" book. $\endgroup$– FrankCommented Sep 18, 2023 at 12:26
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$\begingroup$ I realise I perhaps spoke too soon and that you are asking about quartics only. Inose has a paper "Defining equations..." which I cannot get hold of atm but that claims to prove such surfaces are all birationally quartics. Of course these surfaces will have many classes of square 4 (ie the K3 has many quartic models), but whether this class is very ample needs to be tested on -2-curves and one can hope this is a lattice problem one can unravel from Inose's paper. $\endgroup$– FrankCommented Sep 18, 2023 at 12:44
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$\begingroup$ I know that there are infinitely many non-isomorphic exceptional K3 surfaces. I guess that there may be infinitely many non-isomorphic quartic exceptional K3 surfaces but I cannot give a clean explanation. I am wondering if somebody can. $\endgroup$– BasicsCommented Sep 18, 2023 at 13:15
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