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We report the experimental results with nanosecond-pulse discharges formed in the air gap between a flat electrode and a sharp electrode. The appearance of anode and cathode spots on the electrodes is studied experimentally. It is considered that bright spots on the flat cathode with positive polarity of the sharp electrode are formed due to the explosive electron emission on the cathode and the dynamic displacement current in the gap. It is also shown that with negative polarity of the sharp electrode, bright spots on the flat anode are formed after changing the polarity of the flat electrode due to the discharge oscillatory mode. Under these conditions, the explosive electron emission firstly forms on the sharp cathode. With negative polarity of the sharp electrode of the subnanosecond-pulse pulser, the runaway electron beam current is measured behind the anode foil with a time resolution of no more than 100 ps. [ABSTRACT FROM AUTHOR]