Equality of generic lengths along primitive Brill–Noether components

Determine whether the generic scheme-theoretic length of every maximal irreducible component of the Brill–Noether locus C_p^1={D in C_p : h^0(C,O_C(D))≥2} always equals the generic scheme-theoretic length of its primitive ancestor component in C_m^1, where the ancestor is obtained by removing the fixed part of a general divisor and k≤m≤p.

Background

The paper studies maximal irreducible components of the loci C_p^1 of effective degree-p divisors moving in a pencil and associates each such component with a unique primitive ancestor in a lower-degree locus C_m^1. A comparison of these components is governed by their generic scheme-theoretic lengths.

The monotonicity argument for the Boij–Söderberg coefficients works without generic reducedness if the generic lengths of a component and its primitive ancestor agree. The authors explain that this equality would extend their theorem, but they do not know whether it always holds.