Polarized surfaces of \(\Delta\)-genus 3 and degree 5 (Q1802559)

We try to classify those polarized surfaces \((M,L)\) of \(\Delta\)-genus 3 and degree 5, for which the linear system \(| L|\) has finite base locus and defines a non-birational rational map. Then the surface obtained by blowing up a point of \(M\) is a double covering of a desingularization of a quadric surface. Moreover, we divide these surfaces into four types according to the shape of the fiber containing the exceptional curve. Three of them are fiber spaces over the projective line and the other one is an irrational ruled surface. Conversely, we show the existence of polarized surfaces in each of the four types.