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

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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. 
@article{MZM_2003_73_5_a10,
     author = {N. V. Timofeeva},
     title = {Varieties of {Complete} {Pairs} of {Zero-Dimensional} {Subschemes} of {Lengths} $\ge2$ and $\ge4$ in {Algebraic} {Surfaces}},
     journal = {Matemati\v{c}eskie zametki},
     pages = {743--752},
     publisher = {mathdoc},
     volume = {73},
     number = {5},
     year = {2003},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/MZM_2003_73_5_a10/}
}
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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/