Beilinson's conjecture on K3 surfaces with an involution
Kalyan Banerjee
TL;DR
The paper proves Beilinson's conjecture for certain K3 surfaces $S$ defined over $\bar{\mathbb{Q}}$ equipped with an involution $i$ such that the quotient $S/i$ is isomorphic to $\mathbb{P}^2$ with a genus six branch component and infinitely many rational curves on $S$. The key strategy combines Voisin's observation that $i_*$ acts trivially on the Albanese kernel $A_0(S)$ with a Roitman-type finite-dimensionality analysis to show the relevant degree-zero Chow maps factor through the Albanese and vanish. A central technical step is showing that the image of the correspondence $\Delta_S-\mathrm{Gr}(i)$ is finite-dimensional, achieved via ramification considerations of a ramified double cover and dimension counts of Jacobian fibrations. Under these hypotheses, the Albanese kernel $A_0(S)$ is trivial, establishing Beilinson's conjecture for such $S$ and highlighting the role of ramification and abundant rational curves in countable-field settings.
Abstract
In this note we prove that the Beilinson conjecture holds for certain examples of K3 surfaces over $\bar {\mathbb{Q}}$ equipped with an involution, when the quotient of the surface by the involution is the projective plane branched along a sextic.