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Title: Convergence of the mirror to a rational elliptic surface
Author: Barrott, Lawrence Jack
ISNI:       0000 0004 7651 9714
Awarding Body: University of Cambridge
Current Institution: University of Cambridge
Date of Award: 2018
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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.
Supervisor: Gross, Mark Sponsor: Trinity College ; Cambridge Philosophical Society
Qualification Name: Thesis (Ph.D.) Qualification Level: Doctoral
Keywords: Mirror Symmetry ; Algebraic Geometry ; Mathematics ; Geometry ; Algebra