The generalized Franchetta conjecture for some hyper-Kähler varieties, II

Abstract

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Algebraic cycles and intersections of three quadrics

“…cit. (in the precise form of [14, proposition 2·6]), we find that Let us now check that injects into cohomology. One observes that the class of the diagonal in can not be expressed as a decomposable cycle (for otherwise Y would not have any transcendental cohomology, which is absurd), and so the required injectivity follows from Proposition 3·8.…”

“…We know that the line bundle is very ample on (this follows from [10, proposition 2·3(iii)]). Once more applying [14, proposition 2·6], it follows that where is the restriction of a hyperplane section. Since S has non-zero transcendental cohomology, the injectivity of follows as above.…”

“…We observe that the set-up has the property of [13]: for this was verified in [13], the general case follows by the same argument (3 distinct points on a line impose 3 independent conditions on the linear system). The argument of [13] (in the precise form of [14, proposition 2·6]) then gives an equality …”

“…[11, lemma 2·0·2]; this uses that Y is Fano so that ). To establish the claim, we reason as in [14, Proof of Proposition 2·8]: the MCK decomposition (plus the fact that Y has transcendental cohomology, plus the relation of Lemma 3·15) yields an identity of the form where P is a symmetric polynomial with -coefficients.…”

“…As in [14, section 2·3], let us write , and (this cycle is nothing but the projector on the motive ). Moreover, let us write We now define the -subalgebra this is the image of in cohomology.…”