Ruled Exceptional Surfaces and the Poles of Motivic Zeta Functions

http://dx.doi.org/10.4153/CJM-2007-047-2
Canad. J. Math. 59(2007), 1098-1120

  Published:2007-10-01
 Printed: Oct 2007

In this paper we study ruled surfaces which appear as an exceptional surface in a succession of blowing-ups. In particular we prove that the $e$-invariant of such a ruled exceptional surface $E$ is strictly positive whenever its intersection with the other exceptional surfaces does not contain a fiber (of $E$). This fact immediately enables us to resolve an open problem concerning an intersection configuration on such a ruled exceptional surface consisting of three nonintersecting sections. In the second part of the paper we apply the non-vanishing of $e$ to the study of the poles of the well-known topological, Hodge and motivic zeta functions.