POLARIZED SURFACES OF Δ-GENUS 3 AND DEGREE 5

Abstract

In this paper we try to classify those polarized surfaces (M, L) of Δ-genus 3 and degree 5, for which the linear system | L | has finite base locus and defines a non-birational rational map. Then a surface obtained by the blowing up at a point of M is a double cover of a desigularization 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 is an irrational ruled surface. Conversely, we show the existence of polarized surfaces in each of the four types.