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Jul 25, 2019 at 8:37 comment added ssx @abx You are right.
Jul 25, 2019 at 4:39 comment added abx @Mere Scribe: The OP is asking for smooth surfaces. Schütt examples have rational singularities.
Jul 24, 2019 at 20:29 comment added ssx @abx and $d=5$ arxiv.org/pdf/1308.2525.pdf
Jul 24, 2019 at 19:29 comment added user45397 @abx: Thanks. I have corrected the mistake in the question.
Jul 24, 2019 at 19:28 history edited user45397 CC BY-SA 4.0
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Jul 24, 2019 at 19:21 comment added abx The "restating" doesn't make sense, $H^2(X,\mathbb{Z})$ is a lattice and $H^2(X,\mathcal{O}_X)$ a vector space. As for the first question, it might be true, but it is known only for $d=4$ or $6$ -- in both cases the Fermat surface does the job. You might have a look at this paper.
Jul 24, 2019 at 19:21 comment added ssx mathoverflow.net/questions/153375/… this seems to be an open problem
Jul 24, 2019 at 18:59 history asked user45397 CC BY-SA 4.0