Details
| Original language | English |
|---|---|
| Pages (from-to) | 158-174 |
| Number of pages | 17 |
| Journal | Mathematische Nachrichten |
| Volume | 295 |
| Issue number | 1 |
| Early online date | 14 Dec 2021 |
| Publication status | Published - 31 Jan 2022 |
Abstract
For X a smooth quasi-projective variety and (Formula presented.) its associated Hilbert scheme of n points, we study two canonical Fourier–Mukai transforms (Formula presented.), the one along the structure sheaf and the one along the ideal sheaf of the universal family. For (Formula presented.), we prove that both functors admit a left inverse. This means in particular that both functors are faithful and injective on isomorphism classes of objects. Using another method, we also show in the case of an elliptic curve that the Fourier–Mukai transform along the structure sheaf of the universal family is faithful and injective on isomorphism classes. Furthermore, we prove that the universal family of (Formula presented.) is always flat over X, which implies that the Fourier–Mukai transform along its structure sheaf maps coherent sheaves to coherent sheaves.
ASJC Scopus subject areas
- Mathematics(all)
- General Mathematics
Cite this
- Standard
- Harvard
- Apa
- Vancouver
- BibTeX
- RIS
In: Mathematische Nachrichten, Vol. 295, No. 1, 31.01.2022, p. 158-174.
Research output: Contribution to journal › Article › Research › peer review
}
TY - JOUR
T1 - Some ways to reconstruct a sheaf from its tautological image on a Hilbert scheme of points
AU - Krug, Andreas
AU - Rennemo, Jørgen Vold
N1 - Funding Information: The authors thank Ben Anthes, Pieter Belmans, John Christian Ottem and Sönke Rollenske for helpful discussions and comments. JVR was supported by the Research Council of Norway through the project “Positivity and geometry of higher codimension subvarieties”, project number 250104.
PY - 2022/1/31
Y1 - 2022/1/31
N2 - For X a smooth quasi-projective variety and (Formula presented.) its associated Hilbert scheme of n points, we study two canonical Fourier–Mukai transforms (Formula presented.), the one along the structure sheaf and the one along the ideal sheaf of the universal family. For (Formula presented.), we prove that both functors admit a left inverse. This means in particular that both functors are faithful and injective on isomorphism classes of objects. Using another method, we also show in the case of an elliptic curve that the Fourier–Mukai transform along the structure sheaf of the universal family is faithful and injective on isomorphism classes. Furthermore, we prove that the universal family of (Formula presented.) is always flat over X, which implies that the Fourier–Mukai transform along its structure sheaf maps coherent sheaves to coherent sheaves.
AB - For X a smooth quasi-projective variety and (Formula presented.) its associated Hilbert scheme of n points, we study two canonical Fourier–Mukai transforms (Formula presented.), the one along the structure sheaf and the one along the ideal sheaf of the universal family. For (Formula presented.), we prove that both functors admit a left inverse. This means in particular that both functors are faithful and injective on isomorphism classes of objects. Using another method, we also show in the case of an elliptic curve that the Fourier–Mukai transform along the structure sheaf of the universal family is faithful and injective on isomorphism classes. Furthermore, we prove that the universal family of (Formula presented.) is always flat over X, which implies that the Fourier–Mukai transform along its structure sheaf maps coherent sheaves to coherent sheaves.
UR - http://www.scopus.com/inward/record.url?scp=85121287307&partnerID=8YFLogxK
U2 - 10.1002/mana.201900351
DO - 10.1002/mana.201900351
M3 - Article
AN - SCOPUS:85121287307
VL - 295
SP - 158
EP - 174
JO - Mathematische Nachrichten
JF - Mathematische Nachrichten
SN - 0025-584X
IS - 1
ER -