Varieties of Complete Pairs of Zero-Dimensional Subschemes of Lengths ≥2 and ≥4 in Algebraic Surfaces

We prove that the varieties \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} $$X_{d_1 d_2 } $$ \end{document} of complete pairs of zero-dimensional subschemes of lengths d1≥ 2, d2≥ 4 on a smooth irreducible projective algebraic surface are singular.