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# Ruled Exceptional Surfaces and the Poles of Motivic Zeta Functions

Published:2007-10-01
Printed: Oct 2007
• B. Rodrigues
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## Abstract

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.
 MSC Classifications: 14E15 - Global theory and resolution of singularities [See also 14B05, 32S20, 32S45] 14J26 - Rational and ruled surfaces 14B05 - Singularities [See also 14E15, 14H20, 14J17, 32Sxx, 58Kxx] 14J17 - Singularities [See also 14B05, 14E15] 32S45 - Modifications; resolution of singularities [See also 14E15]