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We obtain a "generalized Franchetta conjecture" type of statement for the Hilbert squares of low genus surfaces, and for the Fano varieties of lines on certain cubic fourfolds.
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Robert Laterveer, The Generalized Franchetta Conjecture for Some Hyperkähler Fourfolds. Publ. Res. Inst. Math. Sci. 55 (2019), no. 4, pp. 859–893