Why ups and downs of amplitude?
A DFT includes an implicit finite length window on any function (which has an infinite domain in the case of a sinusoid).  The effect of multiplication by that window in the time domain on the DFT result is a convolution with the transform of that window.  In the case of a rectangular window, that looks like something very close to a Sinc function (except near DC or Fs/2), or sin(w)/w (which has lots of "ups and downs".)