Two varieties Z and Z are said to be related by extremal transition if there exists a degeneration from Z to a singular variety Z and a crepant resolution Z → Z. In this paper we compare the genus-zero Gromov-Witten theory of toric hypersurfaces related by extremal transitions arising from toric blow-up. We show that the quantum D-module of Z, after analytic continuation and restriction of a parameter, recovers the quantum D-module of Z. The proof provides a geometric explanation for both the analytic continuation and restriction parameter appearing in the theorem. CONTENTS 1. Introduction 1 2. Gromov-Witten theory preliminaries 6 3. Toric preliminaries 15 4. Geometric setup for extremal transitions 21 5. The total spaces 29 6. Crepant transformation conjecture 35 7. D-module of the partial compactification 41 8. Main theorem 46 References 51

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