## Convergence of the mirror to a rational elliptic surface

##### Открыть

##### Автор

Barrott, Lawrence Jack

##### Advisors

Gross, Mark

##### Date

2018-11-30##### Awarding Institution

University of Cambridge

##### Author Affiliation

DPMMS

##### Qualification

Doctor of Philosophy (PhD)

##### Language

English

##### Type

Thesis

##### Metadata

Показать полную информацию##### Citation

Barrott, L. J. (2018). Convergence of the mirror to a rational elliptic surface (Doctoral thesis). https://doi.org/10.17863/CAM.32378

##### Аннотации

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.

##### Keywords

Mirror Symmetry, Algebraic Geometry, Mathematics, Geometry, Algebra

##### Sponsorship

Trinity College Internal Graduate Studentship
Cambridge Philosophical Society Studentship

##### Identifiers

This record's DOI: https://doi.org/10.17863/CAM.32378

##### Rights

Attribution 4.0 International (CC BY 4.0)

Licence URL: https://creativecommons.org/licenses/by/4.0/