MathOverflow

I am studying Dugger's A Primer on Homotopy Colimits, Theorem 18.3, which gives a spectral sequence$$E_2^{p,q} = H^p(I;\mathcal{E}^q(D)) \implies \mathcal{E}^{p+q}\left(\operatorname{hocolim}_{I} D\right)$$for a diagram $D$ over a small category $I$...

Can anyone give me an argument or show me a reference why the product of two (weakly) normal complex analytic spaces is (weakly) normal. This statement appear in Barlet and Magnusson's book complex analytic cycles I p.319 as an exercise. Thank you!

Let $X$ be the K3 surface obtained by resolving the double cover of $\mathbb{P}^2$ branched along the six lines
$$b=0,\quad b=1,\quad c=0,\quad c=1,\quad b+c=0,\quad b+c=1$$
(affine coordinates $b,c$). This arrangement has slopes $\{0,\infty,-1\}$ with unit gaps. Its singularities are 3 ordinary triple points — $\{L_1,L_3,L_5\}$, $\{L_1,L_4,L_6\}$, $\{L_2,L_3,L_6\}$ — and 6 nodes (3 from the parallel pairs at infinity, ...

I need examples of problems that use, directly or indirectly, the homomorphism $S_4\to S_3$ in the solution (its kernel is $\mathbb{Z}_2\oplus\mathbb{Z}_2$).
Obvious candidates:
Lagrange resolvent (the reduction of quartic to cubic equations).