Tillmann-Morris, Hannah;
(2025)
Detecting Blow-ups via Mirror Laurent Polynomials.
Doctoral thesis (Ph.D), UCL (University College London).
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Abstract
The classification of Fano varieties up to deformation is a longstanding open problem. The Fanosearch program is an approach to Fano classification which uses mirror symmetry to translate the geometric classification question into a combinatorial problem. Under mirror symmetry, deformation classes of n-dimensional Fano varieties conjecturally correspond to mutation classes of rigid maximally mutable Laurent polynomials in n variables. In this thesis, we use this correspondence to better understand the birational classification of Fano varieties, by asking the question: Is there a combinatorial condition on pairs of Laurent polynomials that is equivalent to their mirror Fano varieties being related by a blow-up? We introduce a new method of constructing a Fano mirror to a given Laurent polynomial, using constructions from the Gross-Siebert program. Our new construction is more complicated than previous approaches, but is more conceptual and applies in significantly greater generality – in particular, it does not rely on a construction of the Fano as a complete intersection inside a toric variety. In the case that two given Laurent polynomials satisfy a particular combinatorial relationship, both mirror schemes produced by our method can be related by a birational map. We prove that the mirrors to the two Laurent polynomials f = x + y + 1/xy and g = x + y + xy + 1/xy produced by our new construction are related by a birational morphism X_g –> X_f. We show moreover that the restriction of this morphism to the general fibres of the families X_g and X_f gives the blow-up of the projective plane P^2 in a single point.
| Type: | Thesis (Doctoral) |
|---|---|
| Qualification: | Ph.D |
| Title: | Detecting Blow-ups via Mirror Laurent Polynomials |
| Open access status: | An open access version is available from UCL Discovery |
| Language: | English |
| Additional information: | Copyright © The Author 2025. Original content in this thesis is licensed under the terms of the Creative Commons Attribution-NonCommercial 4.0 International (CC BY-NC 4.0) Licence (https://creativecommons.org/licenses/by-nc/4.0/). Any third-party copyright material present remains the property of its respective owner(s) and is licensed under its existing terms. Access may initially be restricted at the author’s request. |
| UCL classification: | UCL UCL > Provost and Vice Provost Offices > UCL BEAMS UCL > Provost and Vice Provost Offices > UCL BEAMS > Faculty of Maths and Physical Sciences UCL > Provost and Vice Provost Offices > UCL BEAMS > Faculty of Maths and Physical Sciences > Dept of Mathematics |
| URI: | https://discovery.ucl.ac.uk/id/eprint/10204135 |
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