I'm reading this paper with the following situation (Beginning of section 2 on page 3):
Let $X$ be a compact Riemann surface of genus $g$ with a finite group $G$ acting on that curve. We write the quotient curve $X/G$ as $X_G$ and the genus of the quotient as $g_0$. Let the cover $X\to X_G$ be branched at $r$ places, $q_1, \cdots, q_r$. The signature of the cover is an $(r+1)$-tuple $$[g_0; s_1, s_2,\cdots , s_r]$$ where the $s_i$ are the ramification indices of the covering at the branch points.
Denoting the map $X\to X_G$ by $f$, by the Riemann-Hurwitz formula $$2(g-1)=2\deg(f)(g_0-1)+\sum_{i=1}^r(s_i-1) $$
Since the left side is even and the degree term is even, it cannot be the case that $r$ is odd with all the $e_i$'s even. But in many tables in the paper, (Eg page 9 where $G$ is taken to be Aut$(X)$ ), there are instances like $$[0; 2,4,6] $$ which should be impossible by the parity argument above. Am I misreading something? If someone could clarify where I am off, I'd be really grateful. The only thing I can think of is that somehow the action of the group is bad so that the quotient map above is not complex analytic to make the Riemann-Hurwitz formula inapplicable. Is this the case?
Whenever we have a map $X\to X_G$, we have to be careful to decide whether we are using ramification points (which live in $X$) or branch points (which live in $X_G$). The Riemann-Hurwitz formula above has a summation over ramification points. There is another version of it using branch points $y_1,\cdots y_k$ with multiplicity $r_i$, which when specialized to a group action as above can be written as
$$2(g(X)-2)=|G|(2g(X_G)-2)+|G|\sum_{i=1}^k(1-\frac{1}{r_i}) $$
which can essentially be derived from the above form by studying the stabilizers of the $G$ action on $X$.
So the example cited above with $|G|=240$ and $g=21$ is actually feasible with signature $[0;2,4,6]$. The mistake the OP (aka I) was making was conflating the notions of ramification points and branch points.