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Howie Nuer : Cubic fourfolds containing some classical surfaces, Kuznetsovs conjecture, and the Kodaira dimension of some C_d



(Special) Cubic fourfolds have garnered a lot of interest lately, most notably because of the difficulty posed by determining their (ir)rationality. We provide explicit descriptions of the generic members of Hassett's divisors C_30, C_38, and C_44 in terms of Fano models of Enriques surfaces and (deformations of) Coble surfaces, and we will use these descriptions to prove the unirationality of these Noether-Lefschetz divisors. After placing this result in the context of a Gritsenko-Hulek-Sankaran-type result and time permitting, we will go on to further enumerate the 13 irreducible components of C_8 \cap C_44 and describe some of them in terms of the rich geometry of Enriques surfaces. In doing so, we describe 7 new components parametrizing cubic fourfolds with trivial Clifford invariant, which are thus rational and verify Kuznetsov's conjecture. Another 6 components have nontrivial Clifford invariant and could provide new nontrivial examples of Kuznetsov's conjecture. This is a work-in-progress.

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