Rank of normal functions and Betti strata

In a recent work of the authors, we proved the generic positivity of the Beilinson-Bloch heights of the Gross-Schoen and Ceresa cycles. The geometric part of the proof was to prove the maximality of the rank of the associated normal function and the Zariski closedness of the Betti strata. In this paper, we generalize these geometric results to an arbitrary family of homologically trivial cycles. More generally, we prove a formula to compute the Betti rank and prove the Zariski closedness of the Betti strata, for any admissible normal function of a variation of Hodge structures of weight $-1$. We also define and prove results about degeneracy loci. In the end, we go back to the arithmetic setting and ask some questions about the rationality of the Betti strata and the torsion loci.