Geodesic
The variety of complete pairs of zero-dimensional subschemes of length 2 of a smooth three-dimensional variety is singular
Sbornik. Mathematics, Tome 194 (2003) no. 3, pp. 361-368 
N. V. Timofeeva. The variety of complete pairs of zero-dimensional subschemes of length 2 of a smooth three-dimensional variety is singular. Sbornik. Mathematics, Tome 194 (2003) no. 3, pp. 361-368. http://geodesic.mathdoc.fr/item/SM_2003_194_3_a2/
@article{SM_2003_194_3_a2,
     author = {N. V. Timofeeva},
     title = {The variety of complete pairs of zero-dimensional subschemes of length 2~of a~smooth three-dimensional variety is singular},
     journal = {Sbornik. Mathematics},
     pages = {361--368},
     year = {2003},
     volume = {194},
     number = {3},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/SM_2003_194_3_a2/}
}
TY  - JOUR
AU  - N. V. Timofeeva
TI  - The variety of complete pairs of zero-dimensional subschemes of length 2 of a smooth three-dimensional variety is singular
JO  - Sbornik. Mathematics
PY  - 2003
SP  - 361
EP  - 368
VL  - 194
IS  - 3
UR  - http://geodesic.mathdoc.fr/item/SM_2003_194_3_a2/
LA  - en
ID  - SM_2003_194_3_a2
ER  - 
%0 Journal Article
%A N. V. Timofeeva
%T The variety of complete pairs of zero-dimensional subschemes of length 2 of a smooth three-dimensional variety is singular
%J Sbornik. Mathematics
%D 2003
%P 361-368
%V 194
%N 3
%U http://geodesic.mathdoc.fr/item/SM_2003_194_3_a2/
%G en
%F SM_2003_194_3_a2

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

Equations are obtained that are satisfied by the vectors of the tangent space to the variety $X_{22}$ of complete pairs of zero-dimensional subschemes of length 2 of a smooth three-dimensional projective algebraic variety at the most special point of the variety $X_{22}$. It is proved that the system of equations obtained is complete and the variety $X_{22}$ is singular.

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

[2] Cheah J., The cohomology of smooth nested Hilbert schemes of points, Thesis, Univ. of Chicago, Chicago, 1994

[3] Timofeeva N. V., “Gladkost i eilerova kharakteristika mnogoobraziya polnykh par $X_{23}$ nulmernykh podskhem dliny 2 i 3 algebraicheskoi poverkhnosti”, Matem. zametki, 67:2 (2000), 276–287 | MR | Zbl

[4] Briançon J., “Description de $\operatorname{Hilb}^n\mathbf C\{x,y\}$”, Invent. Math., 41 (1977), 45–89 | DOI | MR | Zbl

[5] Fogarty J., “Algebraic families on an algebraic surface”, Amer. J. Math., 90 (1968), 511–521 | DOI | MR | Zbl

[6] Kleiman S., “The enumerative theory of singularities”, Real and complex singularities, Proc. Ninth Nordic Summer School/NAVF Sympos. Math. (Oslo, August 5–25, 1976), Sijthoff and Noodhoff, Alpen aan den Rijn, 1977, 297–396 | MR