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Bloch's conjecture and valences of correspondences for K3 surfaces

Bloch's conjecture for a surface $X$ over an algebraically closed field $k$ states that every homologically trivial correspondence $Γ$ acts as 0 on the Albanese kernel $T(X_Ω)$, where $Ω$ is a universal domain containing $k$. Here we prove that, for a complex K3 surface $X$, Bloch's conjecture is equivalent to the existence of a valence for every correspondence. We also give applications of this result to the case of a correspondence associated to an automorphisms of finite order and to the existence of constant cycle curves on $X$. Finally we show that Franchetta's conjecture, as stated by K.O'Grady, holds true for the family of polarized K3 surfacees of genus $g$, if $ 3 \le g \le 6$