his is joint work with Eran Makover, Bjoern Muetzel and Robert Silhol. In a converging family of Riemann surfaces where a separating geodesic is pinched to zero the period matrix converges to the direct sum of the period matrices of the two limit surfaces into which the pinched surface eventually decays. By “period matrix” we understand here the Gram period matrix over the reals.

If the geodesic in the family is non separating the limit is a Riemann surface with cusps, the matrices do not converge and in fact, no period matrix is defined for the limit. In the talk we propose an intrinsically defined infinite family of matrices on the limit surface that may be understood as a blowing up of the missing, or ill defined, period matrix.

We do not know whether the blowing up has applications, but it is a definition. Furthermore, just like period matrices of compact surfaces the members in the blowing up are symplectic albeit not a priori being period matrices themselves.