The variety of complete pairs of zero-dimensional subschemes of an algebraic surface

A. S. Tikhomirov

Izvestiya. Mathematics , Tome 61 (1997) no. 6, pp. 1265-1291

Résumé

We investigate the variety $X$ of zero-dimensional subschemes (that is, systems of points) $Z_1$ and $Z_2$ of given lengths $\deg Z_1=d_1$ and $\deg Z_2=d_2$ on a smooth projective algebraic surface $S$. The variety is realized as the blowing-up of the direct product $\operatorname{Hilb}_{d_1}S\times\operatorname{Hilb}_{d_2}S$ of Hilbert schemes of points along the incidence graph. It is proved that $X$ is naturally isomorphic to the variety of biflags $Z_1\subset Z\supset Z_2$, where $\deg Z=d_1+d_2$. We also study the problem of the smoothness of $X$. It is proved that $X$ is smooth for $d_1=1$ and an arbitrary $d_2\geqslant 1$ using the Kodaira–Spenser rank map in the theory of determinantal varieties and also in the case when $d_1=d_2=2$ by means of a direct geometric consideration.