Reducibility of spectral curves of finite Jacobi pencils

We consider finite pencils of Jacobi matrices $$J_n(w)=A+wB,$$ where $A$ is diagonal and $B$ is tridiagonal with zero diagonal. The spectral curve is the affine plane curve $${\chi }_n(\lambda ,w)=\det (\lambda I+J_n(w))=0.$$ The main question is to describe when this curve is reducible. We prove generic irreducibility for fixed pairwise distinct diagonal entries and discuss several elementary reducibility mechanisms. Besides disconnected Jacobi chains, constant eigenvalue branches, and reflection-symmetric components, one must also take into account reducibility caused by scalar diagonal blocks. We formulate a reducibility conjecture and record low-dimensional evidence and counterexamples to several overly optimistic classifications. A central point of the picture is a codimension-growth principle: apart from the cutting divisors $b_i=0$, genuinely connected primitive reducibility should move to higher and higher codimension as the size of the chain grows.