64 lines on smooth quartic surfaces
  Let k be a field of characteristic other than 2,3. We prove that there are no geometrically smooth quartic surfaces in IP^3 with more than 64 lines. As a key step, we derive the sharp bound that any line meets at most 20 other lines on a smooth quartic. 
Slawomir Rams, Matthias Schuett
math.AG math.NT