Given a line bundle $L$ over a real Riemann surface, we study the number of real zeros of a random section of $L$. We prove a rarefaction result for sections whose number of real zeros deviates from the expected one.

In a real algebraic variety, among all real algebraic hypersurfaces of degree $d$, the proportion of those with a very large total Betti number goes to zero exponentially fast when $d$ goes to infinity.