Nef divisors on the compactified modular curve level $N$ Consider the compactified modular curve with full level structure $X=\overline{\Gamma(N)\setminus \mathcal{H}}$. 
We know the Hodge bundle (the extension of the hodge bundle to the compactification) $L$ is nef and big. If $\partial X=D$ is the boundary divisor we know the canonical divisor to be $K_X=2L-D$.
We know the hodge bundle to be nef. I am looking for some reference on the nef cone of $X$. Specifically I am interested in the question:
For what $k$ is $kL-D$ nef?
 A: For curves, bigness/ampleness/nefness is just a fancy way of asking whether the degree of the divisor is positive. The degree of $K_X$ is $2g_N-2$, and the degree of $D$ is equal to the number $c_N$ of cusps. So the question is equivalent to asking when $k(g_N - 1 + c_N/2) - c_N$ is positive. 
Suppose that $N \ge 3$, to avoid issues with orbifolds. The Galois group $G$ of $X(N)$ over $X(1)$ is
$$G = \mathbf{SL}_2(\mathbf{Z}/N \mathbf{Z})/\{\pm 1\}.$$
Since $X(1)$ has only one cusp, the group $G$ acts transitively on the cusps of $X(N)$. The stabilizer of $\infty$ just consists of the matrices 
$\displaystyle{\left(\begin{matrix} 1 & * \\ 0 & 1 \end{matrix} \right)}$, which is a group of order $N$. Hence, by the orbit-stabilizer theorem,
$$c_N  = \frac{1}{N} \cdot |G|.$$
Finally, by using the Riemann-Hurwitz theorem, the genus of $X(N)$ is
$$g_N = 1 + \frac{1}{12} |G|  - \frac{c}{2}.$$
(We use the fact that there are no elliptic points on $X(N)$, since $N \ge 3$.)
Thus
$$ \deg(k L - D) = k(g_N - 1 + c_N/2)  - c_N =
|G| \left(\frac{k}{12} - \frac{1}{N} \right),$$
which is positive if and only if $k > 12/N$.  
