Geodesic
Varieties of Complete Pairs of Zero-Dimensional Subschemes of Lengths $\ge2$ and $\ge4$ in Algebraic Surfaces
Matematičeskie zametki, Tome 73 (2003) no. 5, pp. 743-752 
N. V. Timofeeva. Varieties of Complete Pairs of Zero-Dimensional Subschemes of Lengths $\ge2$ and $\ge4$ in Algebraic Surfaces. Matematičeskie zametki, Tome 73 (2003) no. 5, pp. 743-752. http://geodesic.mathdoc.fr/item/MZM_2003_73_5_a10/
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     author = {N. V. Timofeeva},
     title = {Varieties of {Complete} {Pairs} of {Zero-Dimensional} {Subschemes} of {Lengths} $\ge2$ and $\ge4$ in {Algebraic} {Surfaces}},
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Voir la notice de l'article provenant de la source Math-Net.Ru

We prove that the varieties $X_{d_1d_2}$ of complete pairs of zero-dimensional subschemes of lengths $d_1\ge2$, $d_2\ge4$ on a smooth irreducible projective algebraic surface are singular.

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