  Let $\mathcal{L}$ be a positive line bundle over a Riemann surface $\Sigma$
defined over $\mathbb{R}$. We prove that sections $s$ of $\mathcal{L}^d$, $d\gg
0$, whose number of real zeros $\#Z_s$ deviates from the expected one are rare.
We also provide asymptotics of the form $\mathbb{E}[(\#Z_s-\mathbb{E}[\#
Z_s])^k]=O(\sqrt{d}^{k-1-\alpha})$ and
${\mathbb{E}[\#Z^k_s]=a_k\sqrt{d}^{k}+b_k\sqrt{d}^{k-1}+O(\sqrt{d}^{k-1-\alpha})}$
for all the (central) moments of the number of real zeros. Here, $\alpha$ is
any number in $(0,1)$, and $a_k$ and $b_k$ are some explicit and positive
constants.Finally, we obtain similar asymptotics for the distribution of
complex zeros of random sections. Our proof involves Bergman kernel estimates
as well as Olver multispaces.

Algebraic Geometry;Probability