Description:
We show that totally real elliptic Lefschetz brations that admit a real section are classi ed by their "real loci" which is nothing but an S1-valued Morse function on the real part of the total space. We assign to each such real locus a certain combinatorial object that we call a necklace diagram. On the one hand, each necklace diagram corresponds to an isomorphism class of a totally real elliptic Lefschetz bration that admits a real section, and on the other hand, it refers to a decomposition of the identity into a product of certain matrices in PSL(2,Z). Using an algorithm to nd such decompositions, we obtain an explicit list of necklace diagrams associated with certain classes of totally real elliptic Lefschetz brations. Moreover, we introduce re nements of necklace diagrams and show that re ned necklace diagrams determine uniquely the isomorphism classes of the totally real elliptic Lefschetz brations which may not have a real section. By means of necklace diagrams we observe some interesting phenomena underlying special feature of real brations.