Let p colon C→ B be a smooth proper family of curves, let L be a line bundle on C, and let pi_n colon C^[n]→ B be the relative Hilbert scheme of length-n subschemes, with tautological bundle L^[n] on C^[n]. Determine universal formulas, valid for every such family and every n≥ 0, for the pushforwards pi_nbigl(s(L^[n])bigr) of the total Segre classes in terms of the relative kappa-classes associated with (C→ B,L). Equivalently, determine all universal coefficients governing the generating series of these Segre-class pushforwards.

Oprea–Pandharipande question on Segre pushforwards and Lehn’s conjecture

Let $p\colon C\to B$ be a smooth proper family of curves, let $L$ be a line bundle on $C$ , and let $\pi_n\colon C^{[n]}\to B$ be the relative Hilbert scheme of length- $n$ subschemes, with tautological bundle $L^{[n]}$ on $C^{[n]}$ . Determine universal formulas, valid for every such family and every $n\ge 0$ , for the pushforwards $\pi_{n*}\bigl(s(L^{[n]})\bigr)$ of the total Segre classes in terms of the relative $\kappa$ -classes associated with $(C\to B,L)$ . Equivalently, determine all universal coefficients governing the generating series of these Segre-class pushforwards.

References

Primary source

arXiv

Additional references

Tautological Pushforwards of Hilbert Schemes of Points on Curves and Surfaces — arXiv

Posted 1 month ago

Progress summary

Refreshed 1 month ago

Claimed solved

A new preprint claims to answer the curve-family question, while covering only one codimension-one part of the broader conjecture.

The Oprea–Pandharipande question concerns universal formulas for Segre-class pushforwards from relative Hilbert schemes. The associated Lehn conjecture has substantial prior results, but the newly claimed result addresses only a codimension-one relative form.

Known results

Marian–Oprea–Pandharipande proved the $K$ -trivial case of Lehn’s conjecture and obtained the $K3$ top Segre formula (2015).
Marian–Oprea–Pandharipande and Szenes–Vergne completed the proof of the absolute Lehn conjecture for nonsingular projective surfaces.
A 2022 result proved the Marian–Oprea–Pandharipande rank-zero Segre-series conjecture for all surfaces.

August 2026 relative pushforwards

A newly reported preprint, Tautological Pushforwards of Hilbert Schemes of Points on Curves and Surfaces, claims relative universality, determines the curve-family coefficients, and thereby answers the stated curve-family question. It also gives formulas for a codimension-one relative form of Lehn’s conjecture; the broader relative conjecture is not claimed to be settled, and the report has not been independently verified.

Current status (as of August 2026): The curve-family question is claimed answered, while only a codimension-one relative form of Lehn’s conjecture is claimed and the broader relative problem remains open.

Sources

No solutions have been posted yet.