I saw a physics demonstration where, between two flat metal plates, was a round conducting ball with large radius. When the two plates had a potential difference, there was an increasing electric field, which increased the electric potential of the conducting ball according to the equation $V = Er$, where $V$ is the electric potential, $E$ is the vector quantity of the electric field, and $r$ is the radius of the sphere. The large electric potential caused a spark across the air.
Then, a spike was placed between the two plates. Electric potential built up in the spike, but because the radius of the "ball" at the end of the spike is small, the electric potential at the end of the spike is small too.
Why does the air not experience dielectric breakdown in the case of the spike? How does current continue to flow through the air, without any sparks?