Algebraic surfaces with Eisenbud–Harris general fibration of genus 4 containing fibers of rank 3

In this paper, we investigate algebraic surfaces with Eisenbud–Harris general fibration of genus 4 satisfying a certain equality with respect to the invariants. In particular, we analyze the structure, prove the existence and construct families for these surfaces. Our surfaces are distinguished into several types from some points of view, namely, certain degenerate fibers and their number, and the direct image of the relative dualizing sheaf for the fibration. Furthermore, we obtain some results for the deformation that are related to these distinctions.