Why are there multiple L-edges in X-ray photoionization? In heavier elements, why are there multiple L-edges in photoionization?
In the image below, what do $\rm{L_{I}}$, $\rm{L_{II}}$ and $\rm{L_{III}}$ stand for?
In this handout, (page 141 Figure 7.13), you can see it for lead (Pb).

 A: Edges are the vertical lines in the graph. As you increase the photon energy, it suddenly becomes possible to excite an electron from the interior of the atom – from an occupied state – and move it to an unoccupied state i.e. to ionize an  atom.
The K,L,M,N... shells are names for the states corresponding to the principal quantum numbers $n=1,2,3,4,\dots$ known from the Hydrogen atom. So the 2 innermost are K, the 2+6 other electrons are L, and so on. The L edge corresponds to the "easiest" ionization kicking out an electron in the L, $n=2$ shell, into an unoccupied state, and so on.
The subscripts $L_I$, $L_{II}$, $L_{III}$ refer to subshells of the L-shell. While in the non-relativistic Hydrogen atom, the energy only depends on $n$, the precise energy of the electrons in a complicated atoms depends on the angular momentum and relative angular momenta, too. So the $l=0$ electrons have slightly different energy than the $l=1$ electrons; and the six $l=1$ electrons are divided to two groups of 2+4 with different $J=1/2$ and $J=3/2$, respectively, which stands for the total spin $J=L+S$. This splitting is due to the spin-orbital interaction. I am not sure what is the ordering of the shells' energies.
I guess it's true in general that the number of subshells is $2n-1=1,3,5,7,\dots$ for $n=1,2,3,\dots$
