We show that a complex planar curve homeomorphic to the projective line has at most four singular points. If it exactly then degree five and is unique up equivalence. Nous montrons qu'une courbe plane complexe homéomorphe à la droite au plus quatre points singuliers. S'il en exactement quatre, alors il est quintique, équivalence près.

Abstract. We prove two conjectures from [DSZ2, DSZ3] concerning the expected number of critical points of random holomorphic sections of a positive line bundle. We show that, on average, the critical points of minimal Morse index are the most plentiful for holomorphic sections of O(N) → CPm and, in an asymptotic sense, for those of line bundles over general Kähler manifolds. We calculate the ex...