Representation-finite Birkhoff type problems for nilpotent linear operators

In the present paper all representation-finite Birkhoff type problems are explicitly described by means of the pairs (I,m). In particular case when I=Ia,b is the union of two incomparable chains I′ and I″ of length |I′|=a−1≥1 and |I″|=b−1≥1, with I′∩I″={⁎}, we study a functorial connection vect-X(p)→Mon(Ia,b,Fm), where vect-X(p) is the vector bundle subcategory of the category coh-X(p) of coherent sheaves over the weighted projective line X(p), for the weight triple p=(a,b,m), with a,b,m≥2, applied by Kussin-Lenzing-Meltzer (2013) [18], in relation with the hypersurface singularity f=x1a+x2b+x3m. In this case we show that Mon(Ia,b,Fm) is representation-finite (resp. representation-tame, representation-wild) if and only if the orbifold Euler characteristic χ(a,b,m)=1a+1b+1m−1 of X(a,b,m) is positive (resp. non-negative, negative).