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We consider mirror symmetry for (essentially arbitrary) hypersurfaces in(possibly noncompact) toric varieties from the perspective of theStrominger-Yau-Zaslow (SYZ) conjecture. Given a hypersurface $H$ in a toricvariety $V$ we construct a Landau-Ginzburg model which is SYZ mirror to theblowup of $V\times\mathbb{C}$ along $H\times 0$, under a positivity assumption.This construction also yields SYZ mirrors to affine conic bundles, as well as aLandau-Ginzburg model which can be naturally viewed as a mirror to $H$. Themain applications concern affine hypersurfaces of general type, for which ourresults provide a geometric basis for various mirror symmetry statements thatappear in the recent literature. We also obtain analogous results for completeintersections.

Lagrangian fibrations on blowups of toric varieties and mirror symmetry for hypersurfaces