In a person with sleep apnea, what causes the rises and falls in oxygen saturation of arterial blood during sleep?

Sleep apnea. Sleep apnea is defined as stopping breathing for 10 seconds. During that time, the body is not getting any oxygen. In addition , there can be instances that last less than 10 seconds, or where a less than adequate effort to breath is attempted. These also reduce the oxygen intake.
No breaths=no oxygen. This is caused by the temporary stoppage/blocking of breathing which is found in sleep apnea. Longer stoppage/blocking of breathing means less oxygen in the blood.
Lack of oxygen. When you stop breathing (apnea) oxygen and carbon dioxide do not exchange in the lungs. Oxygen drops (hypoxia) and co2 increase (hypercapnia)... Classic response in a patient with apnea.