Lines on quartic surfaces
  We show that the maximal number of (real) lines in a (real) nonsingular spatial quartic surface is 64 (respectively, 56). We also give a complete projective classification of all quartics containing more than 52 lines: all such quartics are projectively rigid. Any value not exceeding 52 can appear as the number of lines of an appropriate quartic. 
Alex Degtyarev, Ilia Itenberg, Al\.i S\.inan Sert\"oz
math.AG