On the correspondence with null trace on surfaces

In his work on the $0$-cycles on surfaces Mumford used a method of induced differentials that was proposed by Severi, and introduced a definition of a trace on differential forms. Lopez and Pirola apply this method to the study of correspondences on surfaces. In my talk, I will prove the following result of these two authors: if a smooth surface $S$ of degree $d \geq 5$ in a three-dimensional projective space is given, and $\Gamma$ is a correspondence with null trace of degree $n$ on $X \times S$, then $n \geq d - 2$, and equality holds only if $\Gamma$ is equivalent to one of the three ‘standard’ types of correspondences.