Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: On the universal family of Hilbert schemes of points on a surface   For a smooth quasi-projective surface $X$ and an integer $n\ge 3$, we show
that the universal family $Z^n$ over the Hilbert scheme $\text{Hilb}^{n}(X)$ of
$n$ points has non $\mathbb{Q}$-Gorenstein, rational singularities, and that
the Samuel multiplicity $\mu$ at a closed point on $Z^n$ can be computed in
terms of the dimension of the socle. We also show that $\mu\le n$.

9
The categories of this paper are: nlin
23
The categories of this paper are: solv-int
35
The categories of this paper are: mtrl-th
21
The categories of this paper are: adap-org
1
The categories of this paper are: cs
22
The categories of this paper are: chao-dyn
15
The categories of this paper are: nucl-ex
36
The categories of this paper are: cmp-lg
37
The categories of this paper are: supr-con
4
The categories of this paper are: cond-mat
10
The categories of this paper are: q-bio
14
The categories of this paper are: math-ph
29
The categories of this paper are: funct-an
13
The categories of this paper are: quant-ph
31
The categories of this paper are: ao-sci
20
The categories of this paper are: acc-phys