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PreprintCC BY-NC-ND 4.0arXiv

Chern-Schwartz-MacPherson classes of symmetric powers and applications

  • Barbara Fantechi
  • Lothar Goettsche
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Abstract

The Chern-Schwartz-MacPherson class cSM(X) is defined in the Chow group A*(X) of a scheme X, and generalizes c(TX)∩ [X] if X is smooth. In this paper we independently compute cSM(SnX)∈ A*(SnX) modulo n! torsion; we use this result to compute the pushforward to SnX of cSM(QuotnX(E)) (and in particular cSM(Hilbn X)) when X is smooth and E is a vector bundle on X. Finally, when X is smooth and complete of dimension $3$ we propose a refined version of the the MNOP conjecture that computes the image of the pushforward of [Hilbn X]vir to A0(SnX) and prove it modulo n! torsion when X is Calabi-Yau.

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Cite

BibTeXRIS
BibTeX
@misc{fantechi2026chernschwartzmacpherson,
  title = {{Chern-Schwartz-MacPherson classes of symmetric powers and applications}},
  author = {Barbara Fantechi and Lothar Goettsche},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2610.31826},
  eprint = {2610.31826},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2610.31826},
  note = {CC BY-NC-ND 4.0},
}
RIS
TY  - UNPB
TI  - Chern-Schwartz-MacPherson classes of symmetric powers and applications
AU  - Barbara Fantechi
AU  - Lothar Goettsche
PY  - 2026
DA  - 2026/10/22
PB  - arXiv
DO  - 10.48550/arxiv.2610.31826
UR  - https://arxiv.org/abs/2610.31826
AB  - The Chern-Schwartz-MacPherson class c_SM(X) is defined in the Chow group A_*(X) of a scheme X, and generalizes c(T_X)∩ [X] if X is smooth. In this paper we independently compute c_SM(S^nX)∈ A_*(S^nX) modulo n! torsion; we use this result to compute the pushforward to S^nX of c_SM(Quot^n_X(E)) (and in particular c_SM(Hilb^n X)) when X is smooth and E is a vector bundle on X. Finally, when X is smooth and complete of dimension $3$ we propose a refined version of the the MNOP conjecture that computes the image of the pushforward of [Hilb^n X]^vir to A_0(S^nX) and prove it modulo n! torsion when X is Calabi-Yau.
ER  -