Algebraic geometry
Smooth components on special iterated Hilbert schemes
Comptes Rendus. Mathématique, Volume 360 (2022) no. G5, pp. 425-429.

Let S be a smooth projective surface with p g =q=0. We show how to use derived categorical methods to study the geometry of certain special iterated Hilbert schemes associated to S by showing that they contain a smooth connected component isomorphic to S.

Received:
Revised:
Accepted:
Published online:
DOI: 10.5802/crmath.307
Classification: 14F08, 14J28, 14J29
Reede, Fabian 1

1 Institut für Algebraische Geometrie, Leibniz Universität Hannover, Welfengarten 1, 30167 Hannover, Germany
@article{CRMATH_2022__360_G5_425_0,
     author = {Reede, Fabian},
     title = {Smooth components on special iterated {Hilbert} schemes},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {425--429},
     publisher = {Acad\'emie des sciences, Paris},
     volume = {360},
     number = {G5},
     year = {2022},
     doi = {10.5802/crmath.307},
     language = {en},
     url = {http://www.numdam.org/articles/10.5802/crmath.307/}
}
TY  - JOUR
AU  - Reede, Fabian
TI  - Smooth components on special iterated Hilbert schemes
JO  - Comptes Rendus. Mathématique
PY  - 2022
SP  - 425
EP  - 429
VL  - 360
IS  - G5
PB  - Académie des sciences, Paris
UR  - http://www.numdam.org/articles/10.5802/crmath.307/
DO  - 10.5802/crmath.307
LA  - en
ID  - CRMATH_2022__360_G5_425_0
ER  - 
%0 Journal Article
%A Reede, Fabian
%T Smooth components on special iterated Hilbert schemes
%J Comptes Rendus. Mathématique
%D 2022
%P 425-429
%V 360
%N G5
%I Académie des sciences, Paris
%U http://www.numdam.org/articles/10.5802/crmath.307/
%R 10.5802/crmath.307
%G en
%F CRMATH_2022__360_G5_425_0
Reede, Fabian. Smooth components on special iterated Hilbert schemes. Comptes Rendus. Mathématique, Volume 360 (2022) no. G5, pp. 425-429. doi : 10.5802/crmath.307. http://www.numdam.org/articles/10.5802/crmath.307/

[1] Bauer, Ingrid; Catanese, Fabrizio Some new surfaces with p g =q=0, The Fano Conference. Papers of the conference organized to commemorate the 50th anniversary of the death of Gino Fano (1871–1952), Univ. Torino, Turin (2004), pp. 123-142 | Zbl

[2] Bauer, Ingrid; Catanese, Fabrizio; Pignatelli, Roberto Surfaces of general type with geometric genus zero: a survey, Complex and differential geometry. Conference held at Leibniz Universität Hannover, Germany, September 14–18, 2009 (Springer Monographs in Mathematics), Volume 8, Springer (2011), pp. 1-48 | MR | Zbl

[3] Belmans, Pieter; Fu, Lie; Raedschelders, Theo Hilbert squares: derived categories and deformations, Sel. Math., New Ser., Volume 25 (2019) no. 3, 37, 32 pages | MR | Zbl

[4] Belmans, Pieter; Krug, Andreas Derived categories of (nested) Hilbert schemes (2019) (https://arxiv.org/abs/1909.04321, to appear in Michigan Mathematical Journal)

[5] Belmans, Pieter; Mukhopadhyay, Swarnava Admissible subcategories in derived categories of moduli of vector bundles on curves, Adv. Math., Volume 351 (2019), pp. 653-675 | DOI | MR | Zbl

[6] Catanese, Fabrizio; Göttsche, Lothar d-Very-ample line bundles and embeddings of Hilbert schemes of 0-cycles, Manuscr. Math., Volume 68 (1990) no. 3, pp. 337-341 | DOI | MR | Zbl

[7] Dolgachev, Igor Algebraic surfaces with q=p g =0, Algebraic surfaces (C.I.M.E. Summer School), Volume 76, Springer, 2011, pp. 97-215 | DOI

[8] Fogarty, John Algebraic families on an algebraic surface. II. The Picard scheme of the punctual Hilbert scheme, Am. J. Math., Volume 95 (1973), pp. 660-687 | DOI | MR | Zbl

[9] Fonarev, Anton; Kuznetsov, Alexander G. Derived categories of curves as components of Fano manifolds, J. Lond. Math. Soc., Volume 97 (2018) no. 1, pp. 24-46 | DOI | MR | Zbl

[10] Huybrechts, Daniel Fourier–Mukai transforms in algebraic geometry, Oxford Mathematical Monographs, Clarendon Press; Oxford University Press, 2006 | Zbl

[11] Huybrechts, Daniel; Lehn, Manfred The geometry of moduli spaces of sheaves, Cambridge Mathematical Library, Cambridge University Press, 2010 | DOI | Zbl

[12] Krug, Andreas; Sosna, Pawel On the derived category of the Hilbert scheme of points on an Enriques surface, Sel. Math., New Ser., Volume 21 (2015) no. 4, pp. 1339-1360 | DOI | MR | Zbl

[13] Krug, Andreas; Vold Rennemo, Jørgen Some ways to reconstruct a sheaf from its tautological image on a Hilbert scheme of points (2018) (https://arxiv.org/abs/1808.05931v1, to appear in Mathematische Nachrichten)

[14] Kuznetsov, Alexander G.; Prokhorov, Yuri G.; Shramov, Constantin A. Hilbert schemes of lines and conics and automorphism groups of Fano threefolds, Jpn. J. Math., Volume 13 (2018) no. 1, pp. 109-185 | DOI | MR | Zbl

[15] Lange, Herbert; Newstead, Peter E. On Poincaré bundles of vector bundles on curves, Manuscr. Math., Volume 117 (2005) no. 2, pp. 173-181 | DOI | Zbl

[16] Lee, Kyoung-Seog; Moon, Han-Bom Positivity of the Poincaré bundle on the moduli space of vector bundles and its applications (2021) (https://arxiv.org/abs/2106.04857)

[17] Mumford, David Further pathologies in algebraic geometry, Am. J. Math., Volume 84 (1962), pp. 642-648 | DOI | MR | Zbl

[18] Narasimhan, Mudumbai S. Derived categories of moduli spaces of vector bundles on curves, J. Geom. Phys., Volume 122 (2017), pp. 53-58 | DOI | MR | Zbl

[19] Narasimhan, Mudumbai S. Derived categories of moduli spaces of vector bundles on curves. II, Geometry, algebra, number theory, and their information technology applications (Springer Monographs in Mathematics), Volume 251, Springer, 2018, pp. 375-382 | DOI | MR | Zbl

[20] Reede, Fabian; Zhang, Ziyu Examples of smooth components of moduli spaces of stable sheaves, Manuscr. Math., Volume 165 (2021) no. 3-4, pp. 605-621 | DOI | MR | Zbl

Cited by Sources: