Universal power-series control of virtual surface Quot Segre pushforwards

Determine whether the generating series of virtual Segre pushforwards for tautological classes on relative surface Quot schemes is governed by universal power series in the relative surface \(\kappa\)-classes.

Background

The paper proves a relative universality theorem for tautological pushforwards on Hilbert schemes of points on smooth surface families and applies it to obtain explicit codimension-one formulas for Segre classes of tautological line bundles.

The proposed unresolved problem asks for an analogous universality statement for the virtual Quot schemes Quot⁡X/B(OX⊕N,n)\operatorname{Quot}_{X/B}(O_X^{\oplus N},n), equipped with their standard virtual classes. It would extend the Hilbert-scheme results to the virtual surface Quot setting studied by Oprea–Pandharipande.

References

Oprea--Pandharipande construct virtual surface Quot schemes in Section~2. In this setting, we ask the following parallel question.

For $N\ge1$, let $\pi:X\to B$ be a smooth projective surface family, let $L$ be a line bundle on $X$, and let $\rho_n:\mathsf Q_n=\operatorname{Quot}{X/B} (O_X{\oplus N},n)\to B$, equipped with its standard virtual class $[\mathsf Q_n]{\vir}$. Denote by $L{\mathsf Q}{[n]}$ the tautological $K$-class on $\mathsf Q_n$. Is

\sum_{n\ge0}qn(\rho_n)_\star \left( s\bigl(L_{\mathsf Q}{[n]}\bigr) \cap[\mathsf Q_n]{\vir} \right)

governed by universal power series in the relative surface $\kappa$-classes?

— Tautological Pushforwards of Hilbert Schemes of Points on Curves and Surfaces  (2608.25447 - Kong, 26 Aug 2026) in Question 1.2, immediately after the discussion of the codimension-one analogue of Lehn’s conjecture in Section 1