  We present a comprehensive study of the geometry of Hilbert $p$-adic
eigenvarieties at parallel weight one intersection points of their cuspidal and
Eisenstein loci. The Galois theoretic approach presents genuine difficulties
due to the lack of good deformation theory for pseudo-characters irregular at
$p$ and reflects the rich local geometry at such points. We believe that our
geometric results lead to deeper insight into the arithmetic of Hilbert
automorphic forms and we produce several applications in Iwasawa theory to
support that.
