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Convergence of the mirror to a rational elliptic surface 10.17863/CAM.32378 Barrott, Lawrence Jack Gross, Mark The construction introduced by Gross, Hacking and Keel in [28] allows one to construct a mirror family to (S, D) where S is a smooth rational projective surface and D a certain type of Weil divisor supporting an ample or anti-ample class. To do so one constructs a formal smoothing of a singular variety they call the n-vertex. By arguments of Gross, Hacking and Keel one knows that this construction can be lifted to an algebraic family if the intersection matrix for D is not negative semi-definite. In the case where the intersection matrix is negative definite the smoothing exists in a formal neighbourhood of a union of analytic strata. A proof of both of these is found in [GHK]. In our first project we use these ideas to find explicit formulae for the mirror families to low degree del Pezzo surfaces. In the second project we treat the remaining case of a negative semi-definite intersection matrix, corresponding to S being a rational elliptic surface and D a rational fibre. Using intuition from the first project we prove in the second project that in this case the formal family of their construction lifts to an analytic family. University of Cambridge 2018-11-30 Thesis Doctoral Doctor of Philosophy (PhD) en Trinity College Internal Graduate Studentship Cambridge Philosophical Society Studentship https://www.repository.cam.ac.uk/handle/1810/285007 https://apollo8-f-pro.lib.cam.ac.uk/bitstreams/58c5e578-3a40-483e-8d2c-b9053e5e86a7/download b336c63478e26a516404d7a8eb355d0d https://apollo8-f-pro.lib.cam.ac.uk/bitstreams/84756902-5ae6-4737-bf32-99f9d996c30a/download 87eda9de84448d1f82354d60eee3eb5f https://creativecommons.org/licenses/by/4.0/ Mirror Symmetry Algebraic Geometry Mathematics Geometry Algebra