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Let X_0 be a projective odd-dimensional hypersurface defined over Q with a single node and let Y be a resolution of X_0. Spencer Bloch has recently conjectured an intimate link between the following two real numbers: (i) the height (in the sense of Beilinson and Bloch) of the difference of two rulings on the exceptional quadric on Y; (ii) the height of the limit mixed Hodge structure of any smoothing deformation of X_0. In a series of lectures in 2021, Alexander Beilinson has announced a proof of Bloch's conjecture.
Beilinson's approach is motivic. In this talk, we restrict to the case where X_0 is a nodal curve, and arrive at a refinement of Bloch's conjecture in this case using only rather elementary methods. Our approach appears to be operable in higher dimensions, which might lead to a proof of a refinement of Bloch's conjecture in the future.
As it turns out, the kind of heights as in (ii) can be computed effectively. Via a suitable refinement of Bloch's conjecture, this opens up a way to compute Beilinson-Bloch heights as in (i) effectively, also in higher (co)dimensions. We illustrate this with an example dealing with a nodal cubic threefold.
Based on joint work with Spencer Bloch and Emre Sertöz.*

Freitag, den 22. Juli 2022 um 13:30 Uhr, in INF 205, SR A Freitag, den 22. Juli 2022 at 13:30, in INF 205, SR A

Der Vortrag folgt der Einladung von The lecture takes place at invitation by Dr. Ana María Botero