ON THE CLASSIFICATION OF IRREGULAR SURFACES OF GENERAL TYPE WITH NONBIRATIONAL BICANONICAL MAP

The present paper is devoted to the classification of irregular surfac es of general type with p(g) equal to or greater than 3 and nonbiratio nal bicanonical map. Our main result is that, if S is such a surface a nd if S is minimal with no pencil of curves of genus 2, then S is the symmetric product of a curve of genus 3: and therefore p(g) = q = 3 an d K-2 = 6. Furthermore we obtain some results towards the classificati on of minimal surfaces with p(g) = q = 3. Such surfaces have 6 equal t o or less than K-2 equal to or less than 9, and we show that K-2 = 6 i f and only if S is the symmetric product of a curve of genus 3. We als o classify the minimal surfaces with p(g) = q = 3 with a pencil of cur ves of genus 2, proving in particular that for those one has K-2 = 8.