Geodesic
On Birational Transformations of Hilbert Schemes of an Algebraic Surface
Matematičeskie zametki, Tome 73 (2003) no. 2, pp. 281-294 
A. S. Tikhomirov. On Birational Transformations of Hilbert Schemes of an Algebraic Surface. Matematičeskie zametki, Tome 73 (2003) no. 2, pp. 281-294. http://geodesic.mathdoc.fr/item/MZM_2003_73_2_a12/
@article{MZM_2003_73_2_a12,
     author = {A. S. Tikhomirov},
     title = {On {Birational} {Transformations} of {Hilbert} {Schemes} of an {Algebraic} {Surface}},
     journal = {Matemati\v{c}eskie zametki},
     pages = {281--294},
     year = {2003},
     volume = {73},
     number = {2},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/MZM_2003_73_2_a12/}
}
TY  - JOUR
AU  - A. S. Tikhomirov
TI  - On Birational Transformations of Hilbert Schemes of an Algebraic Surface
JO  - Matematičeskie zametki
PY  - 2003
SP  - 281
EP  - 294
VL  - 73
IS  - 2
UR  - http://geodesic.mathdoc.fr/item/MZM_2003_73_2_a12/
LA  - ru
ID  - MZM_2003_73_2_a12
ER  - 
%0 Journal Article
%A A. S. Tikhomirov
%T On Birational Transformations of Hilbert Schemes of an Algebraic Surface
%J Matematičeskie zametki
%D 2003
%P 281-294
%V 73
%N 2
%U http://geodesic.mathdoc.fr/item/MZM_2003_73_2_a12/
%G ru
%F MZM_2003_73_2_a12

Voir la notice de l'article provenant de la source Math-Net.Ru

We give a precise description of the closure $\Gamma_f$ of the graph of the birational isomorphism $f\colon\operatorname{Hilb}^d\widetilde S\dasharrow\operatorname{Hilb}^dS$ of the Hilbert schemes of points on algebraic surfaces that corresponds to the blow-up $\sigma\colon\widetilde S\to S$ centered at a point on the smooth algebraic surface $S$. We prove that the projection $\operatorname{pr}_{\widetilde H}\colon\Gamma_f\to\widetilde H=\operatorname{Hilb}^d\widetilde S$ is the blow-up centered in the incidence subvariety $R\subset\widetilde H$ that parametrizes $d$-tuples of points in $\widetilde S$ such that at least two of these points are incident to the exceptional line of the blow-up $\sigma$; here $R$ is endowed with a scheme structure by means of a suitable sheaf of Fitting ideals. It is shown that $\Gamma_f$ is smooth only for $d\le2$, and a precise description of the decomposition of the second projection $\operatorname{pr}_H\colon\Gamma_f\to H=\operatorname{Hilb}^dS$ into a composition of two blow-ups with smooth centers in the nontrivial case $d=2$ is given.

[1] Fogarty J., “Algebraic families on an algebraic surface. II: The Picard scheme of the punctual Hilbert scheme”, Amer. J. Math., 95 (1973), 660–687 | DOI | MR | Zbl

[2] Khartskhorn R., Algebraicheskaya geometriya, Mir, M., 1984

[3] Tikhomirov A. S., “Mnogoobrazie polnykh par nulmernykh podskhem algebraicheskoi poverkhnosti”, Izv. RAN. Ser. matem., 61:6 (1997), 153–180 | MR | Zbl

[4] Mamford D., Lektsii o krivykh na algebraicheskoi poverkhnosti, Mir, M., 1968

[5] Tikhomirov S. A., “Punktualnye skhemy Gilberta maloi dliny v razmernostyakh 2 i 3”, Matem. zametki, 67:3 (2000), 414–432 | MR | Zbl

[6] Tikhomirov A. S., “Gladkaya model punktualnykh skhem Gilberta poverkhnosti”, Tr. MIAN, 208, Nauka, M., 1995, 318–334 | MR | Zbl

[7] Fulton U., Teoriya peresechenii, Mir, M., 1989

[8] Shafarevich I. R., Osnovy algebraicheskoi geometrii, Nauka, M., 1988