A primer on Seshadri constants
  Seshadri constants express the so called local positivity of a line bundle on a projective variety. They were introduced by Demailly. The original idea of using them towards a proof of the Fujita conjecture failed but they quickly became a subject of intensive study quite in their own right. Lazarsfeld's book "Positivity in Algebraic Geometry" contains a whole chapter devoted to local positivity and serves as a very enjoyable introduction to Seshadri constants. Since this book has appeared, the subject witnessed quite a bit of development. It is the aim of these notes to give an account of recent progress as well as to discuss many open questions and provide some examples. 
Thomas Bauer, Sandra Di Rocco, Brian Harbourne, Michal Kapustka,   Andreas Leopold Knutsen, Wioletta Syzdek, Tomasz Szemberg