Stein factorization and monodromy of the relative Hilbert scheme of conics

Determine the Stein factorization of the proper relative Hilbert scheme of conics over the open locus of smooth linear sections, characterize its relation to the sheaf of geometric connected components of the open smooth-conic locus, and determine the monodromy acting on those components beyond the low-codimension range.

Background

The paper studies the open Hilbert scheme R_2(Y_E) of smooth conics in general Plücker linear sections and proves uniform irreducibility and rationality only for codimension at most three. In higher codimension, the paper gives examples where connectedness or irreducibility fails and emphasizes that nonminimal components require separate analysis.

The proper relative Hilbert scheme over the parameter space of linear sections provides a natural global object for studying how components of smooth-conic loci vary in families. Its Stein factorization, its comparison with the sheaf of geometric connected components, and the associated monodromy beyond the low-codimension range remain unresolved.

References

Let

\mathcal H_2\longrightarrow Gr(r,\wedgekV\vee)

be the proper relative Hilbert scheme of conics in the universal linear section. What is its Stein factorization over the open locus parametrizing smooth linear sections? How does it relate to the sheaf of geometric connected components of the open smooth-conic locus, and what monodromy acts on those components beyond the low-codimension range?

— Moduli of Conics on General Plucker Linear Sections of Grassmannians  (2609.04727 - Fu, 4 Sep 2026) in Section Further questions, third Question